Front propagation is an ubiquitous feature of reactive phenomena, in particular in the dynamics of excitable media. Standard mathematical counterparts are traveling-waves for reaction-diffusion equations, starting from the Fisher--KPP and the Allen--Cahn equations. These are both of parabolic nature as consequence of a modeling of the diffusion process based on the Fick's law. In this talk, I will discuss models where the diffusion is described by means of a velocity-jump process, leading to a reaction-diffusion equation of hyperbolic type. A basic example, determined by the choice of two possible opposite velocities with equal probability of speed reversal, is called Allen--Cahn equation with relaxation. Results relative to this case, obtained in collaboration with C.Lattanzio (L'Aquila, Italy), R.Plaza (UNAM, Mexico), C.Simeoni (Nice, France), will be presented with emphasis on the numerical exploration of the model. Then, the intention is to discuss possible future research directions for the case of an arbitrary, but finite, number of speeds.