In this paper, we analzye propagating fronts in the context of hyperbolic theories of dissipative processes. These can be considered as a natural alternative to the more classical parabolic models. Emphasis is given toward the numerical computation of the invasion velocity. The first Section is devoted to the presentation of different models for reaction-diffusion phenomena, supporting the idea of the advantages of a description based on hyperbolic equations. Among other advantages, such modeling could provide a detailed description of the transient dynamics of the phenomenon under observation. Three basic numerical schemes are also presented; two of them can, in principle, be applied to general hyperbolic systems, at the price of reduced performances when dealing with discontinuous initial data. In the second Section, we focus on a specific class of 2 × 2 system corresponding to second order partial differential equations in one space dimension, adapted for simplified modeling of reaction-diffusion equations. Specifically, we focus on notable traveling wave solutions, called propagation fronts . Particular cases where the speed of propagation can be explicitly computed are also provided. The third (and final) Section starts with the presentation of the phase-plane algorithm which bears a reliable approximation of the propagation speed, assessing its validity in the case with damping where an explicit formula is available. Then, we propose two PDE-based algorithms to approximate such velocity, named, respectively, scout&spot algorithm (based on tracking the level curve of some intermediate value of the profile) and LeVeque–Yee formula (given by the average value of the discrete transport velocity). Finally, we attest the well-foundedness of both the approaches and conclude by suggesting the second one as more efficient tool in the determination of the speed.