The paper introduces the Hilbert space-valued Wiener process and the corresponding Itô stochastic integral, which are then used to establish existence and uniqueness of weak solutions for linear and semilinear stochastic evolution problems in Hilbert space. The theory is applied to the linear heat and wave equations driven by additive noise. The content covers functional analysis essentials, including spaces of linear operators, pseudo-inverse, and the Cameron-Martin space, as well as elements of Banach space-valued stochastic analysis, such as infinite-dimensional Wiener processes, Wiener processes with respect to a filtration, and martingales in Banach space. The paper also discusses the stochastic integral for nuclear and cylindrical Wiener processes, and provides examples of applying the theory to specific equations.The paper introduces the Hilbert space-valued Wiener process and the corresponding Itô stochastic integral, which are then used to establish existence and uniqueness of weak solutions for linear and semilinear stochastic evolution problems in Hilbert space. The theory is applied to the linear heat and wave equations driven by additive noise. The content covers functional analysis essentials, including spaces of linear operators, pseudo-inverse, and the Cameron-Martin space, as well as elements of Banach space-valued stochastic analysis, such as infinite-dimensional Wiener processes, Wiener processes with respect to a filtration, and martingales in Banach space. The paper also discusses the stochastic integral for nuclear and cylindrical Wiener processes, and provides examples of applying the theory to specific equations.