  We prove a stochastic Taylor expansion for SPDEs and apply this result to
obtain cubature methods, i. e. high order weak approximation schemes for SPDEs,
in the spirit of T. Lyons and N. Victoir. We can prove a high-order weak
convergence for well-defined classes of test functions if the process starts at
sufficiently regular initial values. We can also derive analogous results in
the presence of L\'evy processes of finite type, here the results seem to be
new even in finite dimension. Several numerical examples are added.
