  For a finite group G acting on a smooth projective variety X, we construct
two new G-equivariant rings: first the stringy K-theory of X, and second the
stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct
a new ring called the full orbifold K-theory of Y. For a global quotient
Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra
of the full orbifold K-theory of the the stack Y and is linearly isomorphic to
the ``orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a
different, ``quantum,'' product, which respects the natural group grading. We
prove there is a ring isomorphism, the stringy Chern character, from stringy
K-theory to stringy cohomology, and a ring homomorphism from full orbifold
K-theory to Chen-Ruan orbifold cohomology. These Chern characters satisfy
Grothendieck-Riemann-Roch for etale maps.
  We prove that stringy cohomology is isomorphic to Fantechi and Goettsche's
construction. Since our constructions do not use complex curves, stable maps,
admissible covers, or moduli spaces, our results simplify the definitions of
Fantechi-Goettsche's ring, of Chen-Ruan's orbifold cohomology, and of
Abramovich-Graber-Vistoli's orbifold Chow.
  We conclude by showing that a K-theoretic version of Ruan's Hyper-Kaehler
Resolution Conjecture holds for symmetric products.
  Our results hold both in the algebro-geometric category and in the
topological category for equivariant almost complex manifolds.
