Nonlinear Schrodinger (NLS) equations are used to model physical phenomena such as intense laser beams and ultra-cold matter waves. These waves can be better confined and controlled when using spatially inhomogeneous media. Examples include light propagation in Photonic Crystal Fibers and Bose-Einstein condensate waves inside an external potential. Nonlinear bound state solutions (often called solitary waves or solitons) offer new insight into the salient features of these complex physical systems. This talk will present recent rigorous, asymptotic and computational results on existence and dynamics of such solutions. The bifurcation of bound states from the edge of spectral bands is analyzed in detail. We prove that in the L2-critical case, perturbed bound states with frequency near the band edge do not undergo wave collapse, yet they are nonlinearly unstable. The ensuing dynamics is elucidated using computations of NLS equations.