Moduli of stable sheaves are interesting objects which reflect several properties of underlying spaces. In the theory of algebraic varieties, the canonical bundle is an important invariant. In particular variety with trivial canonical bundle are particulary important. For surfaces, K3 surfaces and abelian surfaces are the examples, and there are many works on these surfaces. Mukai proved that the moduli of stable sheaves on these surfaces are also good by showing they are holomorphic symplectic monifolds. Mukai also invented a quite important tool called Fourier-Mukai transform. With these machinary, he conjectured many interesting phenomena on the moduli spaces in 1980's, and most of them are confirmed affirmatively by Bridgeland's works on Fourier-Mukai transforms and stability conditions. In this lecture, I will explain some results on the moduli of stable sheaves on abelian surfaces including some of Mukai's conjectures: 1. Semi-homogeneous vector bundles and Fourier-Mukai transforms. 2. Stability conditions on abelian surfaces. 3. Moduli of stable sheaves and the Bogomolov factor. 4. Birational invariants and birational automorphisms of moduli spaces.