  Local normal form theorems for smooth equivariant maps between
infinite-dimensional manifolds are established. These normal form results are
new even in finite dimensions. The proof is inspired by the Lyapunov-Schmidt
reduction for dynamical systems and by the Kuranishi method for moduli spaces.
It uses a Slice Theorem for Fr\'echet manifolds as the main technical tool. As
a consequence, the abstract moduli space obtained by factorizing a level set of
the equivariant map with respect to the group action carries the structure of a
Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient
by a compact group of the zero set of a smooth map. The general results are
applied to the moduli space of anti-self-dual instantons, the Seiberg-Witten
moduli space and the moduli space of pseudoholomorphic curves.
