  Multiloop algebras determined by $n$ commuting algebra automorphisms of
finite order are natural generalizations of the classical loop algebras that
are used to realize affine Kac-Moody Lie algebras. In this paper, we obtain
necessary and sufficient conditions for a $Z^n$-graded algebra to be realized
as a multiloop algebra based on a finite dimensional simple algebra over an
algebraically closed field of characteristic 0. We also obtain necessary and
sufficient conditions for two such multiloop algebras to be graded-isomorphic,
up to automorphism of the grading group.
  We prove these facts as consequences of corresponding results for a
generalization of the multiloop construction. This more general setting allows
us to work naturally and conveniently with arbitrary grading groups and
arbitrary base fields.
