Evolution of galaxies has been a concern of astronomers for a long time. Ever since it was realized that stars are not eternal and evolve, people have asked the question how galaxies themselves evolve with a changing star population. The road to our current understanding has been full of cul-de-sacs and false turns. And our current understanding is hardly complete. Over the years of effort to conquer this problem the nature of the question itself has evolved. Initially the question was confined to one of how the colors of a galaxy change with an aging stellar population. Gradually it was realized that this was connected to the evolution of different generations of stars and the chemistry of galaxies. Recently it has been realized that the motions of stars are also subject to evolution. However while the questions have evolved so have observations of the distant galaxies. This has allowed us to investigate the evolution of galaxies sensibly with observational data to focus and drive theoretical arguments.
Sir James Jeans is perhaps best known for arguing for the realist position that the Physical Universe available to us to touch, feel and hear was ``real'' in that it wasn't simply an illusion generated by mental constructs of ideal concepts. Hence it is reasonable to take the empiricist position that it is possible to arrive at an understanding of the Universe by making measurements and observing the Physical Universe rather than introspective meditation. He also suggested that galaxies should be seen as streams of fluid that fill space, gravitating and smoothly flowing. Whatever confidence we may have on his first assertion we have gradually come to realize that his second assertion is not an appropriate way of looking at galaxies. It is more useful to see galaxies as made up of individual stars that are moving in their particular orbits. All the stars contribute collectively to a gravitational background that the stars move in. Once this backdrop, of gravitational pushes and pulls, has been calculated the motion of all the stars can be easily calculated. This completely characterizes the galaxy. At least as far as the description of the motion of all the stars is concerned. This seemingly impossible task is infact simplified by the fact that the origin of the stellar motions is gravitational. This severely restricts the type of motions they are allowed. The trajectories or the orbits of the stars fall into two broad categories. One type of orbit is called the box orbit. These have no net rotation, i.e, the stars in such orbits go around the center of the galaxy as often in one sense (say clockwise) as the other (anti-clockwise). And the orbit can pass through the center of the galaxy. The other type of orbit is called the tube orbits. These are orbits that have net rotation and cannot pass through the center of the galaxy. The fraction of stars in either type of orbit characterizes the galaxy and dictates its shape. Flat, disklike galaxies like the disk of our galaxy are made primarily of stars in tube orbits about the center of the galaxy in the plane of the disk. Elliptical galaxies on the other have a significant fraction of stars in box orbits. As a consequence the elliptical orbits are not as flat or disklike as the disk galaxies. Also because in a box orbit stars pass through the center of the galaxy the centers of elliptical galaxies tend to be dense.
This however isn't quite the caste system of orbits it appears. The fraction of stars in a particular type of orbit can change in a galaxy. If, for example, an elliptical galaxy has a mass concentration in its center, like a black hole or a dense cluster of stars, stars in box orbits can feel very substantial gravitational kicks on their way into the center of the galaxies and move into orbits that are tube orbits. As a consequence the galaxy will become rounder (containing more tube orbits) and also less dense in the center. The reverse can also happen (stars changing from tube to box orbits). But this is probably rarer, although this has been invoked in the models which have elliptical galaxies form by the collision of two or more spiral galaxies. This means that galaxies can change shapes even in the absence of any external perturbation. Naturally in the presence of perturbative forces, like nearby massive galaxies or actual encounters can severely alter the orbits of stars and lead to substantial re-structuring of the galaxy.
Orbits of stars are also deeply connected with the evidence for dark
matter. Of course in disk galaxies the rotation of the stars in tube
orbits that make up the disk appear to be driven by an invisible
material that has lead to the conjecture of dark matter. However there
is another equally strong evidence. You may have noticed that usually
when operating a fire-hose several fire-fighters are needed to hold
onto the end of the hose from which the water is coming out. This is
because left to itself the end will flap around wildly like a
snake. This is referred to as the fire-hose instability. The streaming
of stars around the disk of flat galaxies in their tube orbits can
also have instabilities of these types. Then the disk of the galaxies
are expected to flap around wildly. However the evidence that they
don't do that shows that there are invisible ``fire-fighters'' who
have a tight grip on the disk. Since these ``fire-fighters'' can't be
seen this is another evidence for dark matter around the disk. On the
other hand, many disks are bent. But not in the way of the expected
result of a flapping disk. These bends in the disks are on a grand
scale. Entire disks are warped into 
-sign shapes. Isolated disks
couldn't support such huge bends in themselves. Left to itself the
warp shape would simply wind up very soon and we would be left with a
very thick disk. These warps are evidence that the disks are in the
grip of a powerful vice that is bending it out of shape as well as
keeping it thin. Again the finger prints of dark matter.
All the chemistry, barring some initial chemistry in the early Universe, happens inside stars. Stars are responsible for cosmic cooking today. And here-in lies the fundamental importance of understanding the chemical evolution of galaxies. Because of the intimate connection of stars and chemical evolution, information on the chemical evolution tells us about the stars, both of today and of the distant past. The information is encapsulated into two functions. One is called star formation rate (SFR), which is the number of stars forming per unit time at any given time. The other is the initial mass function (IMF), which describes the fraction of stars formed of a specific mass at any given time. These two functions together describe the star formation in a galaxy at any time. Since this is intimately connected to chemical evolution of a galaxy understanding it will tell us about the history of star formation of that galaxy.
One of the peculiar nomenclature of astronomy is that all elements heavier than helium is referred to as metals. These are a tiny part of most of the Universe. The earth is an anomaly in this respect. Most of the galaxy and the Universe are made up of hydrogen and helium with trace amounts of other elements. The fraction of material in hydrogen is referred to as X, the fraction in helium as Y and the fraction in metals as Z. It is very important to measure Z as this is the measure to which stars have ``polluted'' the pristine material from the Early Universe. This has to be done by taking spectra of the region in question, and measuring the amount of metals present from the absorption or emission features. This, as can be expected, is extremely difficult. Normally a metal that is relatively abundant and has prominent spectral features, like iron is chosen. The relation between iron abundance and Z is calibrated from regions of known Z. Then all we need to do is to measure the abundance of iron of any region and we will have measured Z for the region. This is true provided the region in question doesn't have anomalous amounts of Iron.
Now all the metals are produced in stars, so the amount of metals present in the galaxy is an indication of the amount of processing gas has undergone in stars. The stars form out of clouds of pristine gas (just hydrogen and helium), they undergo nuclear burning inside them all their lives, and then if they are heavy enough (or are in binary systems) explode in supernovae releasing gas enriched with metals to the galactic environment. So some amount of gas is converted into stars, some whom after some time (many millions of years) return some of the gas back to the galaxy, enriched with metals. The simplest model for the chemical evolution of a galaxy then can be a closed box model where we start with some fixed amount of pristine gas that is continuously processed into stars that form at a certain rate. Stars above a certain mass die after some time and return a fixed amount of gas back enriched by a particular fraction of metals. This continues so that at any given time the fraction of metals in the gas of a galaxy is increasing by a quantity that depends on the yield of metals from the star and amount of gas left. Remember the amount of gas left is steadily decreasing because a lot of the stars, those of the mass of sun and lower don't return the gas that was used up in forming them.
Unfortunately this simple model doesn't work for the disk of our galaxy. It predicts that we should see substantial numbers of stars with very low or zero metallicity which we don't see at all in the disk of our galaxy. This is referred to as the G dwarf problem, because long lived G dwarf stars (like our sun) are expected to live longer than the age of our disk so there ought to some around from the beginning of our disk when the gas that they formed from had little metal in them. There are two ways in which the simplest model can be modified to solve this problem. One is to not have a closed box, i.e., to allow gas from outside to fall into the disk. If we add pure hydrogen/helium gas to the disk at about the same rate at which gas is turning into stars, then each generation of stars are forming from roughly the same mixture of hydrogen, helium and metals. And so the metal content of the gas doesn't change substantially with each generation. Another option is to allow a pre-enrichment of the disk. That is instead of starting off with pristine gas we star with gas that was initially polluted with metals, presumably from stars that had formed and exploded before the gas had settled into the disk.
This lead to the ELS model of the formation of the galaxy, proposed by Eggen, Lynden-Bell and Sandage (hence the name). In this model the cloud of gas that was to form the galaxy started off in a (nearly) spherical configuration. While collapsing it forms stars. They explode enriching the gas. This gas collapses further forming stars that continue to enrich the gas. This enriched gas settles into a disk to form stars in it. But because of the pre-enrichment by the spheroidal stars the disk has no star of metallicity less than a particular level. The discovery that the nuclear bulge of our galaxy was very metal rich led to the first modification of the ELS model. The bulge was believed to have formed later from the enriched gas from the disk making it even more metal rich than the disk. This model had the additional advantage of explaining the observed variation of metallicity (the fraction of metals) in galaxies. The metal fraction in the gas is not uniform over the galaxy. But increases from the outside where there is the least metals to the centers of the galaxies where there is the most. The outside-in formation of the Galaxy in the ELS model neatly explains this increasing metallicity as we move in.
But it runs into problems of its own. Even in this model it is expected that there ought to be some stars in the halo of our galaxy that have no or very little metals. Which is not seen. You recognize that we've come back to our G dwarf problem again. All we had succeeded in doing with our modification to the original simple closed box model is that we had pushed back the problem to an earlier epoch of our Galaxy's history. But eventually we must face it. This is because stars are the only place where metals can be produced and there has to be some stars that was the first in our galaxy and consequently has no (or very little) metals in it.
One of the solutions that have been suggested to solve this problem is to propose another generation that came before the present stars. This generation is referred to as the Population III stars. These stars are obviously not seen which needs to be explained. One possible reason could be that they were very large and consequently are all dead now. This is not entirely as ad-hoc as it may seem. Don't forget that hydrogen doesn't form molecules unless the gas has metals which can form dust particles. So the earliest stars must have formed not from cold molecular gas as modern stars do but from warmer atomic gas. This could mean that the earliest stars were systematically more massive than modern stars. But this has not been definitively shown to be the case. Another way in which these population III stars can be hidden is to suggest that their surfaces were covered by gas blown off by exploding massive population III stars like being covered by slime off someone sneezing in front of you. They would then have a layer of gas on their surface which would have more metal in it than the gas making up the star itself.
Another observation is that the disk itself has a varying metallicity. Like the rest of the galaxy, it has the least metals far from the center of the disk and most towards the center. But if the disk form all at once when the gas from the spheroidal cloud settled in a rotating disk how was this gradient in metallicity achieved? Two suggestions have been made to explain this. One is that more stars tend to form towards the center than the outer parts. So the central end of the disk has gone through more generations of stars than the outer parts and hence has more metals. However the observed difference in star formation rates are not as widely different from the inner disk to the outer disk as is needed to explain the metallicity gradient. But because the density of the inner disk is larger than the density in the outer disk the pull of gravity in the inner disk is stronger than in the outer disk. This means that the expanding gas in supernova explosions (which is what enriches the gas) is better confined in the inner disk than in the outer disk. So the central parts of the disk produces more stars and is better able to confine the enriched gas produced in supernovae than the outer parts of the disk is. This could explain the metallicity gradient observed in the disk of the galaxy.
Another issue not unconnected with this problem is one with gas in clusters of galaxies. Our local group is dominated by two average galaxies, our Milky Way and the Andromeda. In contrast to this poverty average clusters of galaxies have thousands of galaxies the size of Milky Way. The centers of these cluster have hundreds of galaxies within a couple of mega parsecs (a million parsecs, remember the separation of Milky Way and Andromeda is about 700 thousand parsecs). But unlike the space between Milky Way and Andromeda which is devoid of gas and stars the space between the galaxies in the cluster is full of gas, extremely tenuous and hot. However this gas is not metal free, on the contrary it is as metal rich as the galaxies themselves! One of the suggestions to explain this phenomenon is to have the galaxies spew out enriched gas through supernovae. However such explosive activity cannot be very good for the galaxies themselves. If the galaxies are expected to fill the inter-cluster region with enriched gas through explosion it is surprising that they haven't blown themselves apart in the process.
All these remain open and unsolved problems in astronomy. In part the improved observations have led to new questions to open up in what was previously solved problems. As more data pours in theories have to be better formulated to be able to explain and are discarded in increasing numbers until the ones that survive are the ones most likely to be the best approximations to reality. We are however still far from this.