The Fukaya category has become a central tool in symplectic topology. In this talk, I will begin with some motivating examples, and explain methods for making computations of the Fukaya categories sufficiently explicit in order to extract information about Lagrangian submanifolds. Mirror symmetry lurks in the background of most of these computations, and the rough idea is that the complexity of the answer is related to the extent to which the mirror deviates from being an honest (commutative) space.