Type Iax supernovae (SN Iax), also called SN 2002cx-like supernovae, are the largest class of peculiar white dwarf (thermonuclear) supernovae, with over fifty members known. SN Iax have lower ejecta velocity and lower luminosities, and these parameters span a much wider range, than normal type Ia supernovae (SN Ia). SN Iax are spectroscopically similar to some SN Ia near maximum light, but are unique among all supernovae in their late-time spectra, which never become fully nebular. SN Iax overwhelmingly occur in late-type host galaxies, implying a relatively young population. The SN Iax 2012Z is the only white dwarf supernova for which a pre-explosion progenitor system has been detected. A variety of models have been proposed, but one leading scenario has emerged: a type Iax supernova may be a pure-deflagration explosion of a carbon-oxygen (or hybrid carbon-oxygen-neon) white dwarf, triggered by helium accretion to the Chandrasekhar mass, that does not necessarily fully disrupt the star.