Stochastic computational modelsare used to simulate complex physical or biological phenomena and to approximate (deterministic) macroscopic physical quantities by means of probabilistic numerical methods.By nature, they often involve singularities and are subject to the curseof dimensionality.Their efficient and accurate simulation is still an open question in many aspects.The aim of this lecture is to review some recent developments concerningthe numerical approximation of singular stochastic dynamics,and to illustrate novel issues in stochastic analysis and PDE analysis thatthey lead to.