  We introduce and discuss L\'evy-type cylindrical martingale problems on
separable reflexive Banach spaces. Our main observations are the following:
Cylindrical martingale problems have a one-to-one relation to weak solutions of
stochastic partial differential equations. Moreover, well-posed problems
possess the strong Markov property and a Cameron-Martin-Girsanov-type formula
holds. As applications, we derive existence and uniqueness results.
