Symmetries of noncommutative spaces and equivariant cohomology

As the title suggests, the main subject of this thesis is the study of symmetries of noncommutative spaces and related equivariant cohomologies. We focus on deformations of classical geometries coming from the action of some symmetry. A close relation between the deformation of the symmetry and the deformation of the space on which it acts is at the heart of our approach; we will use this idea to generate noncommutative geometries, and to de¯ne algebraic models for the equivariant cohomology of such actions.