Take a sphere made of an elastic fabric which is yellow outside and blue inside. This fabric is fantom-like and can auto-intersect (!) but it cannot be pinched neither torn. Can one deform this sphere to put the yellow inside and the blue outside ? Smale proved the existence of such a deformation in 1957. The first explicit recipe is found in 1961 ; for (different) recipes click here and there. I will talk about a recent recipe due to Chéritat. The problem of everting the sphere turns out to be part of a vast family of problems in differential geometry where "everything that is not forbidden is allowed" ; I will illustrate this principle and distribute popcorn during the session.