We can use groups and symmetries to define new statistical models, and to investigate them. In this talk, I will discuss two families of multivariate Gaussian models: 1. RDAG models: graphical models on directed graphs with colored vertices and edges, 2. Gaussian group models: multivariate Gaussian models that are parametrized by a group. I will focus on maximum likelihood estimation, an optimization problem to obtain parameters in the model that best fit observed data. For RDAG models and Gaussian group models, the existence of the maximum likelihood estimate relates to linear algebra conditions and to stability notions from invariant theory. Based on joint work with Carlos Améndola, Kathlén Kohn, Visu Makam, and Philipp Reichenbach.