Klein proposed Group Theory as a means of formulating and understanding geometrical constructions. Geometric Group Theory embraces this approach and also reverses it by using geometrical ideas to give new insights into central problems in Group Theory. In the last decades, it has become a nexus between several branches of mathematics such as Geometry, Model Theory, Dynamical Systems and Algebraic Geometry over Groups.One of the most representative exponents of this interdisciplinary connection is the theory of limit groups. This theory played a crucial role in the recent solution of the famous Tarski problems and revealed a beautiful and deep relation with the theories of JSJ decompositions and very small actions on real trees.As the geometry of free groups is associated to trees, the geometry of partially commutative groups is associated to higher-dimensional analogues of trees. Partially commutative groups are not simply generalisations of free groups, they appear naturally in many different branches of mathematics as well as in computer science, robotics and theoretical physics. This project aims at developing a theory of limit groups over partially commutative groups from algebraic, geometric, algorithmic and model theoretic viewpoints. It intends to explore and strengthen the interconnection between the aforementioned branches of mathematics and to open up directions for further research in each of them.