In the present contribution we discuss the dynamic stability of traveling wave solutions –including constant equilibria– to first order scalar hyperbolic balance laws $$ \partial_tu + \partial_x(f (u)) = g(u), \quad u : \mathbb{R} \times \mathbb{R} \to \mathbb{R} ,$$ where $f$ and $g$ are regular real functions, accounting respectively for advection and reaction processes. Our discussion is based on the recent works of the author and L.M. Rodrigues.