Classical numerical methods for PDEs, like the Finite Element Method, may lead to instabilities in certain challenging problems (e.g. in advection-dominated-diffusion and wave propagation problems), and as a result, we might obtain non-physical solutions. In GEODPG, we focused on designing new computationally efficient and unconditionally stable numerical simulation solvers for approximating time-dependent PDEs. In particular, we developed a class of methods based on the Discontinuos Petrov-Galerkin (DPG) methodology. A robust derivation and an efficient implementation of the DPG method as a time-marching scheme was previously unexplored. During the first two years of the project, we successfully developed efficient and stable solvers for both parabolic and hyperbolic problems, together with adaptive strategies to accurately approximate predefined quantities of interest. Moreover, we developed routines for approximating exponential-related functions of matrices (a key step in our simulations) that are orders of magnitude faster than the current state-of-the-art.