We study the time evolution of the interfaces in one-dimensional bistable extended dynamical systems with discrete time. The dynamics are governed by the competition between a local piecewise affine bistable mapping and any couplings given by the convolution with a function of bounded variation. We prove the existence of travelling wave interfaces, namely fronts, and the uniqueness of the corresponding selected velocity and shape. This selected velocity is shown to be the propagating velocity for any interface, to depend continuously on the couplings and to increase with the symmetry parameter of the local nonlinearity. We apply the results to several examples including discrete and continuous couplings, and the dynamics of planar fronts in multidimensional coupled map lattices. We use our technique to study the existence of other kinds of fronts. Finally we consider a more general class of bistable extended mappings for which the couplings are allowed to be nonlinear and the local map to be smooth.