The paper surveys the past and current uses of groupoids in a wide variety of areas of mathematics, with emphasis on the author’s own interests. The subjects discussed here include: the fundamental groupoid of a space and its use in the computation of the fundamental group of unions of spaces (generalizations of the Van Kampen theorem) and of orbit spaces; results on the category of groupoids and their applications in group theory; the homotopy theory of groupoids; applications to the Galois theory of rings and to ergodic theory; the use of topological groupoids in algebraic geometry and foliation theory; the classifying space of a topological groupoid; and structured groupoids, including multiple groupoids with connections, and their applications in the derivation of higher dimensional variants of the Van Kampen theorem. The paper concludes with some speculations on possible future developments and with a detailed bibliography.