  The solution of a parabolic stochastic partial differential equation (SPDE)
driven by an infinite-dimensional Brownian motion is in general not a
semi-martingale anymore and does in general not satisfy an It\^{o} formula like
the solution of a finite-dimensional stochastic ordinary differential equation
(SODE). In particular, it is not possible to derive stochastic Taylor
expansions as for the solution of a SODE using an iterated application of the
It\^{o} formula. Consequently, until recently, only low order numerical
approximation results for such a SPDE have been available. Here, the fact that
the solution of a SPDE driven by additive noise can be interpreted in the mild
sense with integrals involving the exponential of the dominant linear operator
in the SPDE provides an alternative approach for deriving stochastic Taylor
expansions for the solution of such a SPDE. Essentially, the exponential factor
has a mollifying effect and ensures that all integrals take values in the
Hilbert space under consideration. The iteration of such integrals allows us to
derive stochastic Taylor expansions of arbitrarily high order, which are robust
in the sense that they also hold for other types of driving noise processes
such as fractional Brownian motion. Combinatorial concepts of trees and woods
provide a compact formulation of the Taylor expansions.
