1. Lie semigroups, ordered symmetric spaces and causality -- Causal semisimple symmetric spaces, the geometry and harmonic analysis -- The halfspace method for causal structures on homogeneous manifolds -- Semigroups in foundations of geometry and axiomatic theory of space-time -- On mathematical foundations and physical applications of chronometry -- 2. Invariant cones, Ol'shanskiĭ-semigroups, exponential semigroups -- On the structure of Lie algebras admitting an invariant cone -- Semigroups of Ol'shanskiĭ type -- Lie groups and exponential Lie subsemigroups -- 3. Convexity theorems, representation theory -- Symplectic convexity theorems, Lie semigroups, and unitary representations -- Holomorphic representations of Ol'shanskiĭ semigroups -- 4. Semisimple Lie groups and semigroups -- Control sets and semigroups in semisimple Lie groups -- The asymptotic semigroup of a semisimple Lie group -- 5. Applications: Control -- Applications of the maximum principle to problems in Lie semigroups -- Totally extremal manifolds for optimal control problems -- 6. Applications: Probability -- Lie semigroups and probability: a survey -- List of contributors.