The theory of the staircase avalanche photodiode (APD) is presented and recent results on a new class of APD's with enhanced ratio of ionization coefficients are reviewed. The staircase APD consists of a multistage graded gap structure where only electrons ionize; the entire ionization energy is provided by large conduction band steps (dynodes). A general expression for the excess noise factor F in terms of the number of stages and the multiplication per stage is presented. For high ionization yields per dynode the F factor is near unity independently of the number of stages, implying virtually noise free multiplication at high gain similar to a photomultiplier. This cannot be achieved in a conventional APD at high gain even if one of the ionization coefficients is zero. A comparison between the noise behavior of the staircase APD and that of a phototube is also presented. A microscopic theory of the ionization yield γ is discussed; to obtain a high γ electrons must approach the dynode with an energy in the order of ten times the optical phonon energy. The possible problem of residual hole-initiated ionization is also discussed. Formulas for the electron and hole initiated multiplications are derived; from a measurement of these quantities one can directly obtain the ionization yield and the residual hole ionization coefficient. Experimental and theoretical results on other structures (superlattice, channeling, graded gap APD's) with high \alpha/\beta ratio are also reviewed and design considerations for a long-wavelength multilayer APD are presented.