A Lie group is a smooth manifold whose underlying set of points is equipped with a structure of a group and the multiplication and inverse maps for the group are smooth maps. In other words, it is a group object internal to Diff.
Usually the manifold considered is over complex or real numbers and finite-dimensional (f.d.), but extensions to some other ground fields and infinite-dimensional setting are also very interesting. A real Lie group is compact (or connected, simply connected, etc) if its underlying space is compact (or connected, simply connected, etc).
Every connected f.d. real Lie group is homeomorphic to a product of a compact Lie group and an Euclidean space. Every abelian connected compact f.d. real Lie group is a torus? (a product of circles ).
There is an infinitesimal version of a Lie group, a so-called local Lie group, where the multiplication and the inverse are only partially defined, namely if the arguments of these operations are in a sufficiently small neighborhood of identity. There is a natural equivalence of local Lie groups by means of agreeing (topologically and algebraically) on a smaller neighborhood of the identity. The category of local Lie groups is equivalent to the category of connected simply connected Lie groups.
Sophus Lie has proved several theorems – Lie's three theorems – on the relationship between Lie algebras and Lie groups. So called Lie's third theorem was about the equivalence of the category of f.d. real Lie algebras and local Lie groups. Élie Cartan has extended this to a global integrability theorem called the Cartan–Lie theorem, nowadays after Serre also called Lie’s third theorem.