The discrete nonlinear Schroedinger equation (dNLS) breaks Galilean invariance. Are there discrete solitary traveling waves of dNLS? Numerical simulations and formal analyses go back to the work of M. Peyrard and M.D. Kruskal. A localized state propagating through the lattice excites radiation modes (lattice phonons). The propagating structure slows as it radiates some of its energy away. The structure then stops advancing and is eventually pinned to a fixed lattice site, where it converges to a discrete solitary standing wave. I will describe recent joint work with Michael Jenkinson (Columbia University), where we construct on-site and off-site solitary waves solitary standing waves by bifurcation methods. These are related to the ``Peireles-Nabbaro barrier'', believed to play an important role in the above phenomena.