  We develop a new off-shell formulation for five-dimensional (5D) conformal
supergravity obtained by gauging the 5D superconformal algebra in superspace.
An important property of the conformal superspace introduced is that it reduces
to the superconformal tensor calculus (formulated in the early 2000's) upon
gauging away a number of superfluous fields. On the other hand, a different
gauge fixing reduces our formulation to the SU(2) superspace of
arXiv:0802.3953, which is suitable to describe the most general off-shell
supergravity-matter couplings. Using the conformal superspace approach, we show
how to reproduce practically all off-shell constructions derived so far,
including the supersymmetric extensions of $R^2$ terms, thus demonstrating the
power of our formulation. Furthermore, we construct for the first time a
supersymmetric completion of the Ricci tensor squared term using the standard
Weyl multiplet coupled to an off-shell vector multiplet. In addition, we
present several procedures to generate higher-order off-shell invariants in
supergravity, including higher-derivative ones. The covariant projective
multiplets proposed in arXiv:0802.3953 are lifted to conformal superspace, and
a manifestly superconformal action principle is given. We also introduce
unconstrained prepotentials for the vector multiplet, the ${\cal{O}}(2)$
multiplet (i.e., the linear multiplet without central charge) and
${\cal{O}}(4+n)$ multiplets, with $n=0,1,\dots$ Superform formulations are
given for the BF action and the non-abelian Chern-Simons action. Finally, we
describe locally supersymmetric theories with gauged central charge in
conformal superspace.
