For most of us that fly fish, it is the casting that causes the most "grief". If you cannot cast a fly rod well enough to get the distance you need along with the necessary accuracy, you are doomed to some frustrating days on the water. If you are a trout guy like me, you need to add a little bit more to the equation...increased accuracy and "delicacy". In the trout world, this is often called "presentation" and it is almost always an issue with trout and most certainly is always critical when using a dry fly.
When you fish slow clear waters like spring creeks or the slow meandering stretches of larger trout streams, it is even more critical. Also, in my experience, it
seems the smaller the fly, the more important it is.
Unless you have been chasing trout for a long time like I have, many fly fishers really only think of the traditional "overhead" cast or maybe they include the "roll" cast as well. Without trying to make fly casting more complex than it already is, there are at least two dozen different and specific fly casts. One of the reasons this number is so large is that many of the casts have multiple names as you will see below.
I have tried to put all these various casts into just a few categories. They are "presentation" casts, "saltwater" casts, "distance" casts, "traditional" casts" and "other casts". I have also narrowed down the number by using the purpose/use of the cast to group them...not just the names given to them (i.e. the serpentine cast and the slack line cast are effectively identical in purpose but have acquired different names over time). I have also listed a few casts that do not fit any particular category like spey casting. I have included "mending" fly line (in the "other" category) as it is not casting really casting but it is a very important skill in many of the listed casts.
I have included various links to articles and videos that I have found to be especially good based upon (1) the quality of the instruction and instructor and (2) the quality of the production.