We study in detail two aspects of 2 and 3 dimensional triangulations of the sphere. In the "Dynamics" part, we construct two models in statistical mechanics, on the set of all triangulations of $S^2$ and $S^3$ respectively, and we show that these models behave as glasses, in the sense that these models show a tremendous slowing down of their dynamics once we lower the temperature. We call such models "topological glasses". In the "combinatorics" part, we study the question of counting the number of triangulations of the 3d sphere $S^3$. This is an old and difficult question and even though we do not solve it, we make some very interesting contributions and we find results we strongly believe are a good step towards finding a solution.