If you have problems to get a certain program working, look at the Links. There you will find the most comprehensive emulation sites on the net.
The number of programs that don't run properly under UAE decreases with every new release. If you find a piece of software to require special treatment, please let the developers (the current maintainer is Brian King - see Contacts) know.
Most Kickstart versions work. There has also been success reports about 68020 Kickstarts. Some special A600 ones which apparently rely on an IDE interface or the PCMCIA port being present don't work properly.
In UAE, sprite problems are still an issue. The AGA support is very preliminary right now. Also the emulation of the OCS does not work perfectly (i.e. sprite collisions....), but this will change in the future... :)
A few broken programs require instruction prefetch and/or exception 3 to be emulated (e.g. Shadow of the Beast I, Katakis and Denaris). These can be forced to run by checking "Compatible Mode" in the CPU Settings section of the GUI.