In this thesis we investigate stochastic evolution equations for random fields X: Omega x [0; T] x U -> R, where [0; T] is a time interval, (Omega; F; P) a measure space representing the randomness of the system, and U is typically a domain in Rd (or again a measure space). More precisely, we concentrate on the parabolic situation where A is the generator of an analytic semigroup on Lp(U). We look for mild solutions so that X has values in Lp(U;Lq[0; T]) almost surely under appropriate Lipschitz and linear growth conditions on the nonlinearities. Compared to the classical semigroup approach, which gives X \in Lq([0; T];Lp(U)) almost surely, the order of integration is reversed. We show that this new approach together with a strong Doob and Burkholder-Davis-Gundy inequality leads to strong regularity results in particular for the time variable of the random field X, e.g. pointwise Hölder estimates for the paths, P-almost surely. For less-optimal regularity estimates we only need the relatively mild assumption that the resolvents of A extend uniformly to Lp(U;Lq[0; T]). However, in the maximal regularity case the difficulty of the reversed order of integration in time and space makes extended functional calculi results necessary. ... mehrAs a consequence, we obtain suitable estimates for deterministic and stochastic convolutions. Using Sobolev embedding theorems, we obtain solutions in Lr(Omega;Lp(U;C\alpha[0; T])). In several applications where A is an elliptic operator on a domain in Rd we show that for concrete examples of stochastic partial differential equations our theory leads to stronger results as known in the literature.