We introduce basic concepts of generalized Differential Geometry of Frölicher and diffeological spaces; we consider formal and non-formal pseudodifferential operators in one independent variable, and we use them to build regular Frölicher Lie groups and Lie algebras on which we set up the Kadomtsev-Petviashvili hierarchy. The geometry of our groups allows us to prove smooth versions of the algebraic Mulase factorization of infinite dimensional groups based on formal pseudodifferential operators, and also an Ambrose-Singer theorem for infinite dimensional bundles. Using these tools we sketch proofs of the well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili (KP) hierarchy in a smooth category. We also introduce a version of the KP hierarchy on infinite dimensional groups of series of non-formal pseudodifferential operators and we solve its initial value problem.