A Beauville Surface is a complex surface with every nice geometric property under the sun, and is defined by letting a finite group G act on a product of two curves. This action is made possible by G having a particular kind of generating set known as a Beauville Structure, so naturally we refer to a group that has a Beauville structure as a Beauville Group. Which groups are Beauville groups? If G is a Beauville group then what are its Beauville structures? What do these guys even look like? In this talk we go at least some of the way to answering these questions. Since finite groups vary wildly in their general nature there should hopefully be at least a little of something for (almost) everyone in this talk.