Topological dynamics is the study of qualitative and asymptotic properties of dynamical systems. It is one of the three main modern formulations of abstract dynamical systems, the others being ergodic theory and smooth dynamics. In particular, topological dynamics only requires basic topology and is thus a nice point of entry into the world of dynamical systems. In this course, we will explore topological dynamics and see some connections to number theory. No prior experience with topological dynamics, ergodic theory, or smooth dynamics is assumed.