Proper illumination is a prerequisite for light microscopy in general but this topic is of prime importance when it comes to fluorescence microscopy. The fluorescence intensity of biological samples is usually very low, at least when compared to the intensity of the excitation light. Fluorescence intensity not only depends on loading of the sample with dye, but is determined by the dyes quantum efficiency (QE), and its ability to undergo repeated excitation/emission cycles (photostability). Quantum efficiency (or quantum yield) is the ratio of emitted and absorbed photons. Absorption efficiencies are quantified in terms of the molar extinction coefficient (e). Both QE and (e) are constants under specific environmental conditions. Fluorescence intensity per dye molecule is proportional to the product of (e) and QE. Additionally, only parts of the emitted light are captured by the microscope optics and reach the detector (eye, camera, photomultiplier): a certain light cone, the size of which depends on (the square of) the numerical aperture of the objective.
Taking all this into account, it is easy to understand that powerful light sources are needed in order to generate enough emission for high quality imaging and reproducible data acquisition. Furthermore, the light sources have to be very stable, especially for time-lapse experiments. That is, flickering and oscillations should be negligible on the time scale of the data uptake; any intensity variations should be considerably lower (sub-percent range) than the signal changes expected during the course of the experiment. This is one of the reasons for which DC power supplies are most often used.
Unfortunately there are no light sources with an even spectral distribution over the entire useful wavelength range (near UV - Vis - near IR). Consequently, the right type of light source has to be chosen according to the needs of the different applications.

Apart from lasers and diodes, generally light sources can be classified into incandescent and non-incandescent.

Incandescent light sources
Incandescent sources generate light when a filament, typically tungsten, is heated by electrical energy. They are either under vacuum (your standard light bulb) or filled with an inert gas (nitrogen or a noble gas) or an inert gas combined with a halogen (halogen bulb). Tungsten bulbs are usually not used for fluorescence microscopy because they are UV and blue deficient but they are useful for standard widefield illumination. Their emission spectrum is temperature dependent.

Non-incandescent light sources
Non-incandescent lamps have no filament and are based on electronic discharge in a gas. Most non-incandescent light sources used for microscopy are mercury or xenon arc lamps. They consist of two electrodes sealed under high pressure in a quartz glass bulb. When an electric current is applied and the voltage between the two electrodes is large enough, the gas becomes ionized and begins to conduct electricity. Electrons bridging the gap between the electrodes excite electrons in the gas atoms to higher energy states upon collision. This energy is released as light when the atoms return to ground state. The gas in common fluorescent mercury vapour lamps produces ultraviolet light. The inside of the glass body is coated with phosphor which absorbs this radiation and re-emits it as bright white light.
Arc lamps need to heat up before they reach maximum intensity. They lose efficiency and are more likely to shatter with time and should not be used beyond their rated lifetime.

Mercury burners
Mercury burners are common in microscopy. A drawback is that they produce a very uneven emission spectrum with pronounced peaks in the near UV (365nm), violet (406nm), blue (435nm), green (546nm), and yellow (578nm). For the rest of the useful wavelength range the emission is rather steady but not nearly as intense. Particularly disadvantageous is the absence of a peak around 480nm where many popular dyes and probes such as Calcium Green, GFP and others absorb.

Xenon arc lamps
The other type of frequently used arc lamps are xenon burners which emit much more evenly throughout the entire visible range than mercury burners. However, the intensity falls off in the near UV but is still suitable for the use of UV dyes. Because of this advantage over mercury burners, xenon lamps are used, for example, in scanning monochromators.
As the intensive peaks in the IR indicate, a considerable amount of energy is released as heat. The heat management in xenon arc lamp housings is an important issue. Fluctuations in temperature, as for example caused by unsteady air flow, have an immediate influence on the light intensity and can thus interfere with quantitative analyses.

Mercury-xenon arc lamps
This is a newer type of lamps filled with an optimum mixture of both gases that combines the characteristics of the two. The lines in the UV region are higher in intensity and sharper when compared with conventional mercury lamps. More important for microscopy is the reduced peak-to-peak intensity fluctuation (which still does not reach that of Xenon lamps, however) and the much longer life time (see below).

Brightness
Mercury arc lamps (HBO) for microscopy are available in wattage from about 50 to 200 watts and xenon burners (XBO) from about 75 to 150 watts. However, the simple equation that power stands for brightness doesn't hold here even if the total luminous flux of the lamps rises with the power (not linearly, though). Crucial is rather the brightness per area unit of the part of the arc that is projected into the back aperture of the objective. This is why the size of the arc is an important criterion. For example, the arc of a 150W XBO is nearly 10 times bigger than that of a 75W XBO but the average brightness is nearly three times lower. The spatial luminous intensity distribution of an arc is rather inhomogeneous. The highest brightness is located directly at the tip of the cathode. This part of the arc is also the most stable and is used as point source for coupling the light into the microscope. Understandably, the design of the coupling optics is of great importance and has to be optimised for the lamp that is used.

Durability
Mercury and 75W xenon burners have a specified life of a few hundred hours while 150W XBOs and mercury-xenon lamps last a few thousand hours. Frequent ignition reduces the life time. The spectral emission characteristics may change and the intensity decrease when the rated lifetime is passed, partly because of cathode erosion. Also, the quartz envelope weakens and the low but latent risk of explosion, due to the very high internal gas pressure, rises.

Caution
Arc burners generate extreme heat and should only be changed after they have sufficiently cooled down.
The very high internal gas pressure causes a latent risk of explosion that rises with age. When changing the burner it is highly recommended to wear safety glasses. The lamps should never be ignited outside the lamp housing and should always be stored in their safety containers. The quartz body of the lamps should not be touched with bare hands to prevent etching.
The extreme intensity of the lamps can cause severe damage to the eyes, they should under no circumstances be observed directly when burning.