The talk will start with a short introduction of graphical models. Graphical models combine multivariate statistical models with a representation of conditional independence properties by a mathematical graph. For the special case of purely discrete random variables the use of algebraic statistics, in particular the Diaconis-Sturmfels-Algorithm, within a model selection procedure will be shown. For the purely continuous case ongoing research on the group structure of the parameter space will be presented, which is related to breakdown properties of parameter estimators.