Subsemigroups of finite-dimensional Lie groups that aregenerated by one-parameter semigroups are the subject ofthis book. It covers basic Lie theory for such semigroupsand some closely related topics. These include orderedhomogeneous manifolds, where the order is defined by a fieldof cones, invariant cones in Lie algebras and associatedOl'shanskii semigroups. Applications to representationtheory, symplectic geometry and Hardy spaces are also given.The book is written as an efficient guide for thoseinterested in subsemigroups of Lie groups and theirapplications in various fields of mathematics (see theUser's guide at the end of the Introduction). Since it isessentially self-contained and leads directly to the core ofthe theory, the first part of the book can also serve as anintroduction to the subject.The reader is merely expected to be familiar with the basictheory of Lie groups and Lie algebras.

Lie semigroups and their tangent wedges.- Examples.- Geometry and topology of Lie semigroups.- Ordered homogeneous spaces.- Applications of ordered spaces to Lie semigroups.- Maximal semigroups in groups with cocompact radical.- Invariant Cones and Ol'shanskii semigroups.- Compression semigroups.- Representation theory.- The theory for Sl(2).