In many situations, we are tempted to extend symplectic geometry in order to deal with problems that are not exactly within the scope of classical symplectic geometry, either in infinite dimensions or in the presence of singularities, or both. The usual definitions and constructions are often adapted heuristically on a case-by-case basis, leading to an accumulation of definitions and constructions that are sometimes contradictory, and in any case lack homogeneity. This is an unsatisfactory situation from a conceptual point of view. I will present what diffeology allows as a unifying formal framework and illustrate with examples.