Resonances are widely studied in most areas of engineering and physics, but the approach remains mostly computational or experimental. The reason is that even reduced models of resonant interactions are typically higher dimensional and exhibit great complexity; therefore, they are inaccessible to textbook techniques from dynamical systems theory. This book offers the first systematic exposition of recent analytic results that can be used to understand and predict the global effect of resonances in phase space. The geometric methods discussed here enable one to identify complicated multi-time-scale solution sets and slow-fast chaos in physical problems. The topics include slow and partially slow manifolds, homoclinic and heteroclinic jumping, universal global bifurcations, generalized (\vS)ilnikov orbits and manifolds, disintegration of invariant manifolds near resonances, and high-codimension homoclinic jumping. The main emphasis is on near-integrable dissipative systems, but a separate chapter is devoted to resonance phenomena in Hamiltonian systems. A number of applications are described from the areas of fluid mechanics, rigid body dynamics, chemistry, atmospheric science, and nonlinear optics. In addition, the theory is extended to infinite dimensions to cover resonances in certain nonlinear partial differential equations, such as single and coupled nonlinear Schr(\ddot o)dinger equations. This self-contained monograph should be useful to mathematicians interested in the geometric theory of multi-and infinite-dimensional dynamical systems, as well as to the applied scientist who wishes to analyze resonances in physical problems.