Many Aurivillius phases of ferroelectric complex bismuth oxides are currently known. With the formula Bi2Am-1BmO3m+3, they all consist of Bi2O2 layers interleaved with perovskite-like Am-1BmO3m+1 layers. Their crystal chemistry and their chemical stability in relation to the number, m, of octahedral sheets within the perovskite-like layers and to various cationic substitutions, have been analysed. The possibility of their ordered intergrowth, i.e. the formation of the homologous series Bi4A2n-1B2n+1O6n+9 of mixed-layer structures has been examined and a comprehensive review of their dielectric properties with respect to various chemical substitutions and to different formation processes has been achieved, permitting an outline of their future potential applications.