Abstract: Some of the words in the title are obviously subject to interpretation. For dynamical systems on finite dimensional spaces, one often equates observable events with positive Lebesgue measure sets, and invariant measures that reflect the large-time behaviors of positive Lebesgue measure sets of initial conditions are considered to be of special importance. I will begin by reviewing these concepts for general dynamical systems, describing a simple dynamical picture that one might hope to be true. Reality is a little messier, but a small amount of random noise will bring this picture about. In the second part of my talk I will consider infinite dimensional systems such as semi-flows arising from dissipative evolutionary PDEs, and discuss the extent to which the ideas above can be generalized to infinite dimensions, proposing a notion of ``typical solutions".