This is an overview talk for a module of the Graduate Course on Tame Phenomena Over the Real Field (http://www.fields.utoronto.ca/activities/21-22/tame-real).

Mathematical control theory is the area of application-oriented mathematics that deals with the basic principles underlying the analysis and design of control systems. The fundamental abstract mathematical notion underlying control theory is that of a dynamical system. There is a natural division of dynamical systems into two main types: continuous and discrete. Here we are interested in the former, for which the fundamental underlying mathematical subject is vector field theory. We aim to understand expansions of "tame"---that is, well behaved in some desired sense---structures on the real field by collections of images of solutions of definable vector fields.