  We investigate N=2, D=5 supersymmetry and matter-coupled supergravity
theories in a superconformal context. In a first stage we do not require the
existence of a Lagrangian. Under this assumption, we already find at the level
of rigid supersymmetry, i.e. before coupling to conformal supergravity, more
general matter couplings than have been considered in the literature. For
instance, we construct new vector-tensor multiplet couplings, theories with an
odd number of tensor multiplets, and hypermultiplets whose scalar manifold
geometry is not hyperkaehler.
  Next, we construct rigid superconformal Lagrangians. This requires some extra
ingredients that are not available for all dynamical systems. However, for the
generalizations with tensor multiplets mentioned above, we find corresponding
new actions and scalar potentials. Finally, we extend the supersymmetry to
local superconformal symmetry, making use of the Weyl multiplet. Throughout the
paper, we will indicate the various geometrical concepts that arise, and as an
application we compute the non-vanishing components of the Ricci tensor of
hypercomplex group manifolds. Our results can be used as a starting point to
obtain more general matter-couplings to Poincare supergravity.
