Let be a real separable Hilbert space, an infinitesimal generator of a -semigroup on , a cylindrical -Wiener process on . A stochastic evolution equation
is studied. First, it is shown that existence and uniqueness of a mild solution to (1) may be established by the method of successive approximations under assumptions upon and which are weaker than Lipschitz continuity and were proposed, in the finite-dimensional case, by T. Taniguchi [J. Differ. Equations 96, No. 1, 152-169 (1992; Zbl 0744.34052)]. In the second part of the paper, it is supposed that the semigroup generated by is compact, but the functions and are merely continuous, of a linear growth, and such that pathwise uniqueness holds for (1). It is proven that Euler and Lie-Trotter approximations converge to the solution of (1) in the -norm and that the solution depends continuously on data and on the coefficients. Finally, equations with nonlinear terms , defined only on an open subset of are considered and existence of a solution to (1) is shown under a hypothesis that a suitable sequence of Lyapunov functions exists.