  The notion of quasicrossed product is introduced in the setting of G-graded
quasialgebras, i.e., algebras endowed with a grading by a group G, satisfying a
"quasiassociative" law. The equivalence between quasicrossed products and
quasicrossed systems is explored. It is presented the notion of
graded-bimodules in order to study simple quasicrossed products. Deformed group
algebras are stressed in particular.
