We present a model of soliton propagation in waveguides with quadratic nonlinearity. Criteria for solitons to exist in such waveguides are developed and two example nanowaveguide structures are simulated as proof of concept. Interactions between quadratic solitons and dispersive waves are analyzed, giving predictions closely matching soliton propagation simulations. The example structures are found to support five different regimes of soliton and quasisoliton existence. Pulse propagation in these example waveguides has been simulated confirming the possibility of soliton generation at experimentally accessible powers. Simulations of multisoliton generation, Cherenkov radiation, and quasisolitons with opposite signs of dispersion in the fundamental and second harmonic are also presented here.