A Poincaré space is a topological space satisfying Poincaré duality, as a compact manifold. This book explains how to perform, in the category of Poincaré spaces, a certain number of geometric constructions which are usual in the world of manifolds (surgery, handle techniques, transversality, etc.). These constructions turn out to be useful in solving manifold problems, such as embeddings and group actions. This is the first book to be published on this subject. The methods are based on standard surgery theory and elementary homotopy arguments.