The observations and theory of hydrogen-deficient Type I supernovae are reviewed. Classical Type Ia supernovae are thought to arise in the thermonuclear explosions of degenerate stellar cores, but Types Ib and Ic may result from core collapse in more massive stars. Binary evolutionary schemes for the origin of SN Ia are described. Calculations of theoretical spectra of SN Ia at various stages in their evolution are described in some detail and the importance of SN Ia for nucleosynthesis and for determination of the cosmological distance scale is summarized. SN Ib are characterized by the appearance of helium lines near maximum and emission lines of oxygen and calcium at later epochs. Like SN Ib, SN Ic lack the evidence for Si near maximum that characterizes SN Ia, but they have only weak indications of He. SN Ib and Ic may arise in the hydrogen-denuded cores of massive stars which may have lost their hydrogen envelopes to a binary companion. The possible connection of SN Ib and Ic events to the 'transition' event, SN 1987K, is also discussed. An appendix describes the operation of the supernova atmosphere code used to calculate early-time spectra of SN I.