  We construct chiral N=(1,0) self-dual supergravity in Euclidean
eight-dimensions with reduced holonomy Spin(7), including all the higher-order
interactions in a closed form. We first establish the non-chiral N=(1,1)
superspace supergravity in eight-dimensions with SO(8) holonomy without
self-duality, as the foundation of the formulation. In order to make the whole
computation simple, and the generalized self-duality compatible with
supersymmetry, we adopt a particular set of superspace constraints similar to
the one originally developed in ten-dimensional superspace. The intrinsic
properties of octonionic structure constants make local supersymmetry,
generalized self-duality condition, and reduced holonomy Spin(7) all consistent
with each other.
