  A Lie groupoid can be thought of as a generalization of a Lie group in which
the multiplication is only defined for certain pairs of elements. From another
perspective, Lie groupoids can be regarded as manifolds endowed with a type of
action codifying internal and external symmetries. The vigorous development of
their theory in the last few decades has been largely stimulated by their
connections with such areas as Poisson geometry and non-commutative geometry,
as well as several topics in mathematical physics, including classical
mechanics, quantization and topological field theories. This article is an
overview on Lie groupoids, including basic definitions, key examples and
constructions, and topics such as actions and representations, local models,
Morita equivalence and cohomology.
