  As a step towards formulating projective superspace techniques for
supergravity theories with eight supercharges, this work is devoted to field
theory in five-dimensional N = 1 anti-de Sitter superspace AdS^{5|8} =
SU(2,2|1)/SO(4,1) x U(1) which is a maximally symmetric curved background. We
develop the differential geometry of AdS^{5|8} and describe its isometries in
terms of Killing supervectors. Various off-shell supermultiplets in AdS^{5|8} x
S^2 are defined, and supersymmetric actions are constructed both in harmonic
and projective superspace approaches. Several families of supersymmetric
theories are presented including nonlinear sigma-models, Chern-Simons theories
and vector-tensor dynamical systems. Using a suitable coset representative, we
make use of the coset construction to develop an explicit realization for one
half of the superspace AdS^{5|8} as a trivial fiber bundle with fibers
isomorophic to four-dimensional Minkowski superspace.
