According to Turing, a real number x is computable if there exists an algorithm which, upon input n, produces a rational number at distance no more than 2^-n from x.  This notion has been extended to treat computability of infinite objects over general metric spaces, and given rise to the theory of Computable Analysis. In this talk we will introduce these abstract notions of computability,  and will illustrate them via some examples taken from  the theory of dynamical systems: invariant sets, invariant measures and generic points.