Theories of particle acceleration in localized wave fields explicitly excluding extended wave fields are reviewed. There are several scenarios including localized fields. The first is stationary double layers. Measurements in the auroral zone do not support the conventional view that large amplitude (strong) double layers exist. If double layers accelerate particles to high energies one requires a multitude of double layers organized in chains along the field in order to build up sufficiently large potentials. This applies also to solar coronal conditions. Caviton radiation is another candidate. High-frequency wave turbulence consisting of Langmuir cavitons may generate power law electron spectra but is unable to accelerate ions. Low-frequency candidates are lower-hybrid wave cavitons and magnetosonic solitary waves. Single lower-hybrid cavitons may accelerate electrons during their transition time up to about 16 times their thermal energy and thus may provide an injection mechanism for further acceleration. Such acceleration may be found in caviton turbulence which has a considerably harder spectrum than extended wave turbulence and may thus be more efficient. Ion acceleration takes place in chaotic interaction of the ions with transverse electric wave fields. Chaotic interaction is also important for electrons showing that the usual theory of quasilinear diffusion coefficients breaks down for wave amplitudes exceeding a certain threshold.