The paper contains two main parts. In the first part, we develop a general theory of crystalline cohomology for algebraic stacks. We study in the stack theoretic setting generalizations of the basic definitions and results of crystalline cohomology, the correspondence between crystals and modules with integrable connection, the Cartier isomorphism and Cartier descent, Ogus1 generalization of Mazurs Theorem, as well as a stack—theoretic generalization of the cle Rham-Witt complex.