I will discuss how Markov chains in infinite dimensions generically have typically have properties which make their ergodic theory difficult. Such properties are very pathological in finite dimensions, but in some sense generic in infinite dimensions. I will draw examples from stochastically forced PDEs and stochastic delay equations. We will see that in infinite dimensions, a typical system acts much more like an hypo-elliptic diffusion then an elliptic one. I will also discuss the existence of spectral gaps as well as the uniqueness of invariant measures. If time permits I will discuss an extension of Hormander's "sum of squares theorem" to infinite dimensions.