Abstract: I will talk about the theoretical challenges to the numerical study of dynamical systems. I will broadly discuss what practitioners attempt to compute, and whether such computations are always possible. Such questions lead to interesting mathematics with surprising practical implications. As an instructive example of the limitations on our ability to compute things, I will describe a "nice" one-dimensional dynamical system for which a numerical approximation of the long-term statistical behavior of the orbits is not possible. In particular, the Monte Carlo simulation provably fails for it.