Using a special form of Ulam's method, we estimate the measure-theoretic entropy of a triple , where M is a smooth manifold, T is a uniformly hyperbolic map, and is the unique physical measure of T. With a few additional calculations, we also obtain numerical estimates of (i) the physical measure , (ii) the Lyapunov exponents of T with respect to , (iii) the rate of decay of correlations for with respect to test functions, and (iv) the rate of escape (for repellors). Four main situations are considered: T is everywhere expanding, T is everywhere hyperbolic (Anosov), T is hyperbolic on an attracting invariant set (axiom A attractor), and T is hyperbolic on a non-attracting invariant set (axiom A non-attractor/repellor).