  We calculate the relevant Spencer cohomology of the minimal Poincar\'e
superalgebra in 5 spacetime dimensions and use it to define Killing spinors via
a connection on the spinor bundle of a 5-dimensional lorentzian spin manifold.
We give a definition of bosonic backgrounds in terms of this data. By imposing
constraints on the curvature of the spinor connection, we recover the field
equations of minimal (ungauged) 5-dimensional supergravity, but also find a set
of field equations for an $\mathfrak{sp}(1)$-valued one-form which we interpret
as the bosonic data of a class of rigid supersymmetric theories on curved
backgrounds. We define the Killing superalgebra of bosonic backgrounds and show
that their existence is implied by the field equations. The maximally
supersymmetric backgrounds are characterised and their Killing superalgebras
are explicitly described as filtered deformations of the Poincar\'e
superalgebra.
