The homological mirror symmetry program studies equivalences between the category of coherent sheaves on a Kähler manifold X and the Fukaya category of a dual Kähler manifold X^. The latter category has historically been difficult to compute. In this talk, we describe how recent developments in microlocal sheaf theory have made calculations in this category possible, and we give a proof (from joint work with Vivek Shende) of homological mirror symmetry for X^ an affine hypersurface. Depending on time and audience interests, we discuss some other examples of calculations in Fukaya categories and explain potential applications to birational toric geometry.