Homomorphisms of abelian varieties over finite fields

These are notes of my lectures at the summer school ``Higher-dimensional geometry over finite fields" in Goettingen, June--July 2007. We present a proof of Tate's theorem on homomorphisms of abelian varieties over finite fields (including the $\ell=p$ case) that is based on a "quaternion trick". In fact, a a slightly stronger version of those theorems with "finite coefficients" is proven.