Integrable 1+1 Dimensional Gravity Models

Integrable models of dilaton gravity coupled to electromagnetic and scalar matter fields in dimensions 1+1 and 0+1 are reviewed. The 1+1 dimensional integrable models are either solved in terms of explicit quadratures or reduced to the classically integrable Liouville equation. The 0+1 dimensional integrable models emerge as sectors in generally non integrable 1+1 dimensional models and can be solved in terms of explicit quadratures. The Hamiltonian formulation and the problem of quantizing are briefly discussed. Applications to gravity in any space time dimension are outlined and a generalization of the so called ‘no hair’ theorem is proven using local properties of the Lagrange equations for a rather general 1+1 dimensional dilaton gravity coupled to matter.