I will talk about a new cohomology theory for algebraic varieties in positive characteristic, called edged crystalline cohomology. This is a generalisation of crystalline cohomology and depends on the choice of a "decay-function''. Linear decay-functions correspond to integral versions of rigid cohomology, while logarithmic decay-functions produce the conjectured family of log-decay crystalline cohomology theories, parametrised by positive real numbers. During the talk, I will explain the construction of this theory after a brief recall of the classical crystalline and rigid cohomology theories.