A class of space-time stochastic processes that arise as solutions of stochastic evolution equations is discussed. By analogy with Itô stochastic differential equations, a stochastic evolution equation is given by the formal equation ∂X ∂t = AX + f(X)W where A is a positive self-adjoint operator and W is a space-time white noise. Existence and continuity results are obtained for equations of the foregoing type when f is a Lipschitz mapping. In addition, an equation is obtained for the covariance function of the solution.