This talk will present the concept of conjugation space or manifold, as introduced by Hausmann, Holm, and Puppe. Roughly speaking it is a space X equipped with an action of the cyclic group of order 2 such that the mod 2 cohomology of the fixed points is abstractly isomorphic to that of X, but all degrees are divided by 2. One should think of complex projective spaces or complex Grassmannians with complex conjugation. The precise definition makes it more rigid than what it looks like and I will present general properties such equivariant spaces enjoy, as well as examples. In joint work with Wolfgang Pitsch we studied concrete geometric examples known as the Floyd manifolds and obtained, with Nicolas Ricka, a characterization by using methods in genuine equivariant stable homotopy theory. I might be able to say a word about this at the end of the talk.