It has long been known that interacting nonlinear waves in dissipative systems can form interesting, globally extended patterns that have a variety of simple symmetries. Here we present new experiments which show that interacting nonlinear waves on the surface of an excited fluid can yield a surprising variety of new nonlinear states which range from highly localized ``solitary modes” and oscillons to spatially extended quasicrystalline states and exotic superlattice states. Both the characteristics of these intriguing nonlinear states and the basic mechanisms leading to their formation will be reviewed.