At the end of the last century, differential geometry has been challenged by theoretical physics. New objects have been displaced from the periphery of the classical theory to the center of attention: irrational tori, quotients of the 2- dimensional torus by irrational lines, with the problem of quasi-periodic potential; orbifolds with the problem of singular symplectic reduction; structures on the infinite dimensional space of connections of principal bundles, in Yang-Mills fields theory; groups and subgroups of symplectomorphisms in symplectic geometry or geometric quantization; coadjoint orbits of groups of diffeomorphisms, the famous Virasoro group, for example. All these objects, belonging to the outskirts of the realm of differential geometry, claimed their place inside the theory, as full citizens. Diffeology gives them satisfaction in a unified framework, bringing simple answers to simple problems, by being the right balance between rigor and simplicity, and pushing off the boundary of classical geometry to include seamlessly these objects in the heart of its concerns.
However, we must note that diffeology didn't spring up on an empty battlefield. Many solutions to these questions have been already proposed, from functional analysis to non commutative geometry, via smooth structures a la Sikorski or a` la Froölicher. For what concerns us, each of these attempts has a flaw: functional analysis is often an overkilling heavy machinery; physicists run fast, if we want to stay close to them we need to jog lightly. Non commutative geometry is uncomfortable for the geometer not familiar enough with the C*-algebra world, where he loses his intuition and sensibility. Sikorski or Froölicher spaces miss the singular quotients. And perhaps most frustrating, none of these approaches embraces the variety of situations at a time.

Patrick Iglesias-Zemmour
Jerusalem 2005-Aix en Provence 2012 Why this book ?