These notes, which are based on a graduate course and seminars given at Wesleyan University in the fall of 1997, contain many of the main results of chromatic and density Ramsey theory in Z. This subfield of combinatorial number theory deserves attention due in part to the fact that its principal theorems are (i) natural, (ii) easy to formulate, and (iii) non-trivial. These three features are hallmarks of the best in pure mathematics. Most of the results we treat were originally proved combinatorially. We concentrate here on a variety of alternative approaches proceeding by way of topological dynamics and ergodic theory. Our goal is not to prove theorems of the greatest generality in the shortest amount of time, but to allow the reader to observe the methodology gradually unfolding.