  Vasiliev's higher-spin theories in various dimensions are uniformly
represented as a simple system of equations. These equations and their gauge
invariances are based on two superalgebras and have a transparent algebraic
meaning. For a given higher-spin theory these algebras can be inferred from the
vacuum higher-spin symmetries. The proposed system of equations admits a
concise AKSZ formulation. We also discuss novel higher-spin systems including
partially-massless and massive fields in AdS, as well as conformal and massless
off-shell fields.
