In this thesis we study stochastic evolution equations in Banach spaces. We restrict ourselves to the two following cases. First, we consider equations in which the drift is a closed linear operator that depends on time and is random. Such equations occur as mathematical models in for instance mathematical finance and filtration theory. Second, we restrict ourselves to UMD Banach spaces with type 2. As the theory of Ito stochastic integration is insufficient for studying equations of this general type, we need to have a proper understanding of several extensions to the Ito integral. Two of such extensions that are considered rigorously in this thesis are the Skorohod integral and the forward integral. Moreover, in Chapter 5, a new solution concept is introduced. The relationship between other solution concepts is discussed. Finally, we prove existence, uniqueness and regularity of solutions to stochastic evolution equations with adapted drift.