In this text, we give an introduction to equivariant cohomology of a manifold. In the case of an $S^1$-action with isolated fixed points, we describe, after localization, the equivariant cohomology of a manifold in terms of fixed points with the help of Paradan's formula. As a consequence, we give a simple proof of the localization formula of Atiyah-Bott-Berline-Vergne.