After reviewing the main concepts for numerical approximations of solutions to stochastic partial differential equations, I will present recent works (and works in progress) on the design and analysis of numerical schemes for the approximation of invariant distributions of (parabolic) SPDEs. I will mainly focus on gradient systems driven by space-time white noise, which admit Gibbs distributions as invariant distributions. For such systems, one is able 1) to approximate the invariant distribution in total variation distance 2) to construct higher-order schemes using preconditioning and/or postprocessing techniques