Perfectoid spaces are a class of mathematical objects in the field of algebraic geometry and number theory. They are defined over a perfect field, which is a type of field where every element has a unique p-th root for a prime number p. Perfectoid spaces help in studying the properties of schemes and provide a framework for understanding various phenomena in arithmetic geometry. These spaces were introduced by mathematician Peter Scholze and have applications in areas such as p-adic Hodge theory and the study of modular forms. Perfectoid spaces allow mathematicians to work with infinite-dimensional objects in a controlled way, leading to new insights and results in modern mathematics.