Deterministic partial differential equations (PDEs) originated from mathematical models for physics and biology. They gradually developed from specific problems (such as heat conduction) to a deep and complex mathematical theory. Stochastic partial equations have been studied since the 60s to extend the range of applicability both of ordinary stochastic equations (Itô equations) and of the classical PDEs. Among notable applications we may list turbulent flows in fluid mechanics and diffusions in random media. The book is exclusively devoted to a study of stochastic PDEs for a random evolution in Hilbert and Banach spaces. Recall that the first existence result harks back to V. V. Baklan (1963). The book summarizes basic facts about stochastic analysis and ordinary stochastic differential equations. It employs eigenfunction expansions, Green functions and Fourier transforms in a study of stochastic transport and heat-wave equations. Specific examples provide a motivation for the investigation of stochastic evolution equations in Hilbert space; general theorems both on the existence and uniqueness and asymptotic behaviour of solutions are proved. The text may be characterized as an excellent guide to current research topics that opens possibilities for further developments in the field.