This book focuses on exact construction of moving excitations in nonlinear dissipative lattice systems. The lattice structure arises from the interaction among units discretely distributed in space where each unit follows a nonlinear evolution dynamics. The class of systems we consider includes Nagumo and FitzHugh-Nagumo (FHN) type pde's. In each of these we replace the commonly considered cubic nonlinearity with an appropriate piecewise linear one. This reduces the problem of exact construction of the moving excitations to one of constructing the general solution of a linear evolution equation in terms of the eigenmodes in appropriate intervals of the relevant propagation variable and a matching of the solutions across those intervals. As solutions of discrete Nagumo model we construct two classes of fronts: kink and antikink, which are related through a symmetry transformation. The solutions to discrete FHN systems are the pulse and pulse-trains constructed on the basis of singular perturbation technique where a conjugate pair of kink and antikink form the leading and trailing edges of the pulse, the two being joined up through the slow evolution of a recovery variable.