Superthermal energetic particles (EP) often drive shear Alfv́n waves unstable in magnetically confined plasmas. These instabilities constitute a fascinating nonlinear system where fluid and kinetic nonlinearities can appear on an equal footing. In addition to basic science, Alfv́n instabilities are of practical importance, as the expulsion of energetic particles can damage the walls of a confinement device. Because of rapid dispersion, shear Alfv́n waves that are part of the continuous spectrum are rarely destabilized. However, because the index of refraction is periodic in toroidally confined plasmas, gaps appear in the continuous spectrum. At spatial locations where the radial group velocity vanishes, weakly damped discrete modes appear in these gaps. These eigenmodes are of two types. One type is associated with frequency crossings of counterpropagating waves; the toroidal Alfv́n eigenmode is a prominent example. The second type is associated with an extremum of the continuous spectrum; the reversed shear Alfv́n eigenmode is an example of this type. In addition to these normal modes of the background plasma, when the energetic particle pressure is very large, energetic particle modes that adopt the frequency of the energetic particle population occur. Alfv́n instabilities of all three types occur in every toroidal magnetic confinement device with an intense energetic particle population. The energetic particles are most conveniently described by their constants of motion. Resonances occur between the orbital frequencies of the energetic particles and the wave phase velocity. If the wave resonance with the energetic particle population occurs where the gradient with respect to a constant of motion is inverted, the particles transfer energy to the wave, promoting instability. In a tokamak, the spatial gradient drive associated with inversion of the toroidal canonical angular momentum P is most important. Once a mode is driven unstable, a wide variety of nonlinear dynamics is observed, ranging from steady modes that gradually saturate, to bursting behavior reminiscent of relaxation oscillations, to rapid frequency chirping. An analogy to the classic one-dimensional problem of electrostatic plasma waves explains much of this phenomenology. EP transport can be convective, as when the wave scatters the particle across a topological boundary into a loss cone, or diffusive, which occurs when islands overlap in the orbital phase space. Despite a solid qualitative understanding of possible transport mechanisms, quantitative calculations using measured mode amplitudes currently underestimate the observed fast-ion transport. Experimentally, detailed identification of nonlinear mechanisms is in its infancy. Beyond validation of theoretical models, the future of the field lies in the development of control tools. These may exploit EP instabilities for beneficial purposes, such as favorably modifying the current profile, or use modest amounts of power to govern the nonlinear dynamics in order to avoid catastrophic bursts. © 2008 American Institute of Physics.