The notion of groupoid is a common generalization of the concepts of space and group. In the theory of groupoids, spaces and groups are treated on equal footing. Simplifying somewhat, one could say that a groupoid is a mixture of a space and a group; it has space-like and group-like properties that interact in a delicate way.

In fact, though groups are sufficient to characterize the symmetries of homogeneous structures, there are plenty of objects (like finite parts of a crystal) which exhibit what we clearly recognize as symmetry, but admit few automorphisms. The correct way to describe such symmetries is to use groupoids.

Also, if we want to classify certain mathematical objects (say Riemann surfaces or Platonic solids) without losing track of their symmetries, we need to use groupoids. In this example the groupoid appears as a refinement of the concept of moduli space, which parametrizes the objects to be classified, but in a way incorporating the symmetries of these objects as well.

The theory of Lie groupoids is one approach to the problem of endowing an abstract groupoid with geometric structure. The theory of stacks is another such approach.

Groupoids were introduced by Brandt in 1926 in a paper on composition of quadratic forms in four variables. Ehresmann added further structures to groupoids and used them as a tool in differential topology and geometry. In analysis, Mackey used groupoids under the name of virtual groups to allow the treatment of ergodic actions of groups and observed that the convolution operation extended from groups to groupoids so that one can construct many interesting non-commutative algebras. In this context, the use of groupoid convolution algebras as substitute for algebras of functions on badly behaved quotient spaces plays a central role in the non-commutative geometry theory of Connes. It provides a unified study of operator algebras, foliations, and index theory. In algebraic topology, the fundamental groupoid of a topological space has been exploited by Higgins, Brown and others.

Grothendiek introduced stacks initially to give geometric meaning to higher non-commutative cohomology classes. This is also the context that gerbes appeared in first. During the 1960s, Artin, Deligne and Mumford worked out the basics of the theory of algebraic stacks, which has been so successful in the theory of moduli. Even though the theory of stacks lay almost dormant for many years, the last decade has seen an explosion of activity. Intersection theory on stacks was pioneered by Gillet and Vistoli. Later the work of Kontsevich and Manin on Quantum Cohomology required a highly sophisticated intersection theory machinery on algebraic stacks. The localization formula for virtual fundamental classes plays a fundamental role in Givental's proof of the mirror conjecture. Kontsevich's recent work on mirror symmetry revealed deep, and as yet poorly understood connections between quantization, deformation theory and stacks.

The theory of Lie groupoids has undergone rapid development in the last fifteen years. It played a fundamental role in the recent development of Poisson geometry. Poisson manifolds can be considered in a certain sense as non-linear generalizations of Lie algebras. The search for group-like objects for Poisson manifolds lead to the discovery of symplectic groupoids by Karasev and Weinstein. Symplectic groupoids are expected to play an essential role in quantization of Poisson manifolds, just like Lie groups do for Lie algebras.

Recently, considerable attention has been directed at the Poisson sigma model of Schaller, Strobl and Ikeda, who studied non-linear gauge theory. Roughly speaking, this is a generalization of usual gauge theory where the gauge group is replaced by a Lie groupoid (in this case the symplectic groupoid of the Poisson manifold). The Poisson sigma model was employed by Cattaneo and Felder in depth, and they showed that the perturbative quantization of this model yields Kontsevich's celebrated star-products.