  We consider the damped hyperbolic equation in one space dimension $\epsilon
u_{tt} + u_t = u_{xx} + F(u)$, where $\epsilon$ is a positive, not necessarily
small parameter. We assume that $F(0)=F(1)=0$ and that $F$ is concave on the
interval $[0,1]$. Under these assumptions, our equation has a continuous family
of monotone propagating fronts (or travelling waves) indexed by the speed
parameter $c \ge c_*$. Using energy estimates, we first show that the
travelling waves are locally stable with respect to perturbations in a weighted
Sobolev space. Then, under additional assumptions on the non-linearity, we
obtain global stability results using a suitable version of the hyperbolic
Maximum Principle. Finally, in the critical case $c = c_*$, we use self-similar
variables to compute the exact asymptotic behavior of the perturbations as $t
\to +\infty$. In particular, setting $\epsilon = 0$, we recover several
stability results for the travelling waves of the corresponding parabolic
equation.
