  The aim of this review paper is to explain the relevance of Lie groupoids and
Lie algebroids to both physicists and noncommutative geometers. Groupoids
generalize groups, spaces, group actions, and equivalence relations. This last
aspect dominates in noncommutative geometry, where groupoids provide the basic
tool to desingularize pathological quotient spaces. In physics, however, the
main role of groupoids is to provide a unified description of internal and
external symmetries. What is shared by noncommutative geometry and physics is
the importance of Connes's idea of associating a C*-algebra C*(G) to a Lie
groupoid G: in noncommutative geometry C*(G) replaces a given singular quotient
space by an appropriate noncommutative space, whereas in physics it gives the
algebra of observables of a quantum system whose symmetries are encoded by G.
Moreover, Connes's map G -> C*(G) has a classical analogue G -> A*(G) in
symplectic geometry due to Weinstein, which defines the Poisson manifold of the
corresponding classical system as the dual of the so-called Lie algebroid A(G)
of the Lie groupoid G, an object generalizing both Lie algebras and tangent
bundles. This will also lead into symplectic groupoids and the conjectural
functoriality of quantization.
