  The solutions of parabolic and hyperbolic stochastic partial differential
equations (SPDEs) driven by an infinite dimensional Brownian motion, which is a
martingale, are in general not semi-martingales any more and therefore do not
satisfy an It\^o formula like the solutions of finite dimensional stochastic
differential equations (SODEs). In particular, it is not possible to derive
stochastic Taylor expansions as for the solutions of SODEs using an iterated
application of the It\^o formula. However, in this article we introduce Taylor
expansions of solutions of SPDEs via an alternative approach, which avoids the
need of an It\^o formula. The main idea behind these Taylor expansions is to
use first classical Taylor expansions for the nonlinear coefficients of the
SPDE and then to insert recursively the mild presentation of the solution of
the SPDE. The iteration of this idea allows us to derive stochastic Taylor
expansions of arbitrarily high order. Combinatorial concepts of trees and woods
provide a compact formulation of the Taylor expansions.
