The classical modelling of diffusion processes posits that the flux is proportional to the gradient of the density, with the final output that the resulting equation is parabolic. Such description has a number of flaws, the most famous being the infinite speed of propagation. The main problem lies in the fact that the system's response is assumed to be instantaneous. Differently, the presence of a time-lag, corresponding to the presence of inertial terms, has to be taken into account. In fact, philosophically speaking, parabolicity is an exceptional event, and the generic case (which preserves well-posedness) is the hyperbolic one. In this presentation, attention is given to the Maxwell-Cattaneo law -a constitutive equation for the evolution of the flux- which is the simplest mechanism to incorporate the presence of a relaxation time-scale and furnishing a hyperbolic model. Specifically, the talk concentrates on the main differences when the standard balance equation is coupled with the Maxwell-Cattaneo law in place of the Fourier law. Specific attention will be given to the presence of propagating fronts and their (analytical and numerical) stability.