A simple dynamical system, which is easy to implement numerically, can nevertheless exhibit chaotic dynamics. Trying to compute the behaviour of a trajectory of such a system for an extended period of time is impractical: small computational errors are magnified very rapidly. The modern paradigm of the numerical study of chaos is the following: since the simulation of an individual orbit for an extended period of time does not make a practical sense, one should study the limit set (attractor) of a typical orbit. From the theoretical computability point of view, the principal problem becomes: suppose that a dynamical system with a single attractor can be numerically simulated. Can its attractor be effectively computed? I will present our joint work with C. Rojas, which gives surprising answers to this question for some of the simplest non-linear dynamical systems. The talk will be self-contained, in particular, no prior knowledge of computability theory will be assumed.