  Supergravity with eight supercharges in a four-dimensional Euclidean space is
constructed at the full non-linear level by performing an off-shell time-like
reduction of five-dimensional supergravity. The resulting four-dimensional
theory is realized off-shell with the Weyl, vector and tensor supermultiplets
and a corresponding multiplet calculus. Hypermultiplets are included as well,
but they are themselves only realized with on-shell supersymmetry. The
off-shell reduction leads to a full understanding of the Euclidean theory. A
complete multiplet calculus is presented along the lines of the Minkowskian
theory. Unlike in Minkowski space, chiral and anti-chiral multiplets are real
and supersymmetric actions are generally unbounded from below. Precisely as in
the Minkowski case, where one has different formulations of Poincar\'e
supergravity by introducing different compensating supermultiplets, one can
also obtain different versions of Euclidean supergravity.
