We analyze linear propagation in negative index materials by starting from a dispersion relation and by deriving the underlying partial differential equation. Transfer functions for propagation are derived in temporal and spatial frequency domains for unidirectional baseband and modulated pulse propagation, as well as for beam propagation. Gaussian beam propagation is analyzed and reconciled with the ray transfer matrix approach as applied to propagation in negative index materials. Nonlinear extensions of the linear partial differential equation are made by incorporating quadratic and cubic terms, and baseband and envelope solitary wave solutions are determined. The conditions for envelope solitary wave solutions are compared with those for the standard nonlinear Schrodinger equation in a positive index material.