A rigid-analytic space is an analytic variety over a non-archimedean field, e.g. the p-adic rationals. Pioneered by Tate in the early 1960's in order to construct uniformizations of certain elliptic curves, the theory has found many striking applications in arithmetic and algebraic geometry. I will discuss the basic theory of rigid spaces and explain their relationship to schemes, Berkovich spaces, and formal geometry. Speaker(s): Matt Stevenson (UM)