We will consider various aspects of structural Ramsey theory in a countable first-order structure, leading to an investigation of several notions of largeness. For ultrahomogeneous (i.e. Fraisse) structures, the Ramsey theoretic properties of the structure and the dynamical properties of the automorphism group are closely related. This talk should serve as an introduction to the Kechris-Pestov-Todorčević correspondence while also discussing directions for new research.