This chapter examines perfectoid spaces. A Huber ring R is Tate if it contains a topologically nilpotent unit; such elements are called pseudo-uniformizers. One can more generally define when an analytic Huber ring is perfectoid. There are also notions of integral perfectoid rings which are not analytic. In this course, the perfectoid rings are all Tate. It would have been possible to proceed with the more general definition of perfectoid ring as a kind of analytic Huber ring. However, being analytic is critical for the purposes of the course. The chapter then looks at tilting and sousperfectoid rings. The class of sousperfectoid rings has good stability properties.