  We survey on-shell and off-shell higher derivative supergravities in
dimensions $1\le D\le 11$. Various approaches to their construction, including
the Noether procedure, (harmonic) superspace, superform method, superconformal
tensor calculus, $S$-matrix and dimensional reduction, are summarized.
Primarily the bosonic parts of the invariants and the supertransformations of
the fermionic fields are provided. The process of going on-shell, solutions to
the Killing spinor equations, typical supersymmetric solutions, and the role of
duality symmetries in the context of $R^4, D^4 R^4$ and $D^6 R^4$ invariants
are reviewed.
