Stellar systems - star clusters, galaxies, dark matter haloes, and so on - are ubiquitous characters in the evolutionary tale of our Universe. This tutorial article is an introduction to the collective dynamical evolution of the very large numbers of stars and/or other self-gravitating objects that comprise such systems, i.e. their kinetic theory. We begin by introducing the basic phenomenology of stellar systems, and explaining why and when we must develop a kinetic theory that transcends the traditional two-body relaxation picture of Chandrasekhar. We study the orbits that comprise stellar systems, how those orbits are modified by perturbations, how a system responds self-consistently to fluctuations in its gravitational potential, and how one can predict the long term fate of a stellar system in various dynamical regimes. Though our treatment is necessarily mathematical, we develop the formalism only to the extent that it facilitates real calculations. We give many examples throughout the text of the equations being applied to topics of major astrophysical importance. Furthermore, in the 1960s and 1970s the kinetic theory of stellar systems was a fledgling subject which developed in tandem with the kinetic theory of plasmas. However, the two fields have long since diverged. Yet once one has become fluent in both Plasmaish and Galacticese, and has a dictionary relating the two, one can pull ideas directly from one field to solve a problem in the other. Therefore, another aim of this tutorial article is to provide our plasma colleagues with a jargon-light understanding of the key properties of stellar systems, to point out the many direct analogies between stellar- and plasma-kinetic calculations, and ultimately to convince them that stellar dynamics and plasma kinetics are, in a deep and beautiful and useful sense, the same thing.