Stochastic partial differential equations (SPDEs, for short) constitute a wide and quickly growing area of research within mathematics, combining the fields of stochastic analysis and partial differential equations (PDEs, for short). In this thesis we study specific questions concerning the discretization and approximation of linear parabolic SPDEs and-intimately connected with it-the regularity of the solutions. SPDEs of parabolic type or stochastic evolution equations are usually treated from an abstract point of view as ordinary stochastic differential equations (SDEs, for short) in an infinite-dimensional state space.