Many multiphysics applications require high-order, physically consistent and computationally efficient discretizations of hyperbolic PDEs. In this talk, we will present a new mass-matrix-free finite element (MF-FEM) scheme, which provides an explicit and arbitrary high order approximation of the smooth solutions of the hyperbolic PDEs both in space and time. The design of the scheme allows for an efficient diagonalization of the mass matrix without any loss of accuracy. This is achieved by coupling the FEM formulation [1] with a Deferred Correction (DeC) type method [2] for the discretization in time. The advantage of such a matrix-free approach consists in preserving a compact approximation stencil even at high orders, which reduces the computational cost compared to classical finite element techniques and provides potential benefit for exascale computing on future computer architectures. We have assessed our method on several challenging benchmark problems for one- and two- dimensional Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions. We will discuss how structure-preserving properties of the proposed MF-FEM method can be enforced using convex limiting for blending the high-order and low-order element residuals [3]. Finally, we will show that the proposed approach extends seamlessly to general advection-diffusion systems, so that the explicit time update in advection-dominated problems can remain matrix-free. Implicit time discretization of the diffusion and stiff sources is coupled with explicit DeC for the advection terms via multirate time integration method [4].