We introduce polynomial jump-diffusions taking values in an arbitrary Banach space via their infinitesimal generator. We obtain two representations of the (conditional) moments in terms of solution of systems of ODEs. We illustrate the wide applicability of these formulas by analyzing several state spaces. We start by the finite dimensional setting, where we recover the well-known moment formulas. We then study the probability-measure valued setting, where we also obtain an existence result for the corresponding martingale problems. Moving to more recent results, we consider (potentially rough) forward variance polynomial models and we illustrate how to use the moment formulas to compute prices of VIX options. Finally, we show that the signature process of a d-dimensional Brownian motion is polynomial and derive its expected value via the polynomial approach. This is in fact just an illustrative example: the same applies to solutions of every SDE with analytic coefficients.