The aim of this text is to present a thorough examination of weakly nonlocal solitary waves, which are just as important in applications as their classical counterparts. The book describes a class of waves which radiate away from the core of the disturbance but are nevertheless very long-lived nonlinear disturbances. Specific examples are provided in the areas of water waves, particle physics, meteorology, oceanography, fiber optics pulses and dynamical systems theory. For many species of nonlocal solitary waves the radiation is exponentially small in 1/E where E is a perturbation parameter, thus lying beyond-all-orders . A second theme is the description of hyperasymptotic perturbation theory and other extensions of standard perturbation methods. These methods have been developed for the computation of exponentially small corrections to asymptotic series. A third theme involves the use of Chebyshev and Fourier numerical methods to compute solitary waves. Special emphasis is given to steadily-translating coherent structures, a difficult numerical problem. A fourth theme is the description of a large number of non-soliton problems in quantum physics, hydrodynamics, instability theory and others where beyond-all-order corrections arise and where the perturbative and numerical methods described earlier are essential. Later chapters provide a thorough examination of matched asymptotic expansions in the complex plane, the small denominator problem in Poincare-Linstead ( Stokes ) expansions, multiple scale expansions in powers of the hyperbolic secant and tangent functions and hyperasymptotic perturbation theory.