  Inspired by the commutator and anticommutator algebras derived from algebras
graded by groups, we introduce noncommutatively graded algebras. We generalize
various classical graded results to the noncommutatively graded situation
concerning identity elements, inverses, existence of limits and colimits and
adjointness of certain functors. In the particular instance of noncommutatively
graded Lie algebras, we establish the existence of universal graded enveloping
algebras and we show a graded version of the Poincar\'e-Birkhoff-Witt theorem.
