The paper contains a revised, and extended by new results, part of the author's PhD thesis. The main objects that we study are toric varieties naturally associated to special Markov processes on trees. Such Markov processes can be defined by a tree T and a group G. They are called group-based models. The main, but not unique, motivation to consider these processes comes from phylogenetics. We study the geometry, defining equations and combinatorial description of the associated toric varieties. We obtain new results for a large class of not necessarily abelian group-based models, which we call G-models. We also prove that equations of degree 4 define the projective scheme representing the 3-Kimura model.