  We describe the supersymmetric completion of several curvature-squared
invariants for ${\cal N}=(1,0)$ supergravity in six dimensions. The
construction of the invariants is based on a close interplay between
superconformal tensor calculus and recently developed superspace techniques to
study general off-shell supergravity-matter couplings. In the case of minimal
off-shell Poincar\'e supergravity based on the dilaton-Weyl multiplet coupled
to a linear multiplet as a conformal compensator, we describe off-shell
supersymmetric completions for all the three possible purely gravitational
curvature-squared terms in six dimensions: Riemann, Ricci, and scalar curvature
squared. A linear combination of these invariants describes the off-shell
completion of the Gauss-Bonnet term, recently presented in arXiv:1706.09330. We
study properties of the Einstein-Gauss-Bonnet supergravity, which plays a
central role in the effective low-energy description of
$\alpha^\prime$-corrected string theory compactified to six dimensions,
including a detailed analysis of the spectrum about the ${\rm AdS}_3\times {\rm
S}^3$ solution. We also present a novel locally superconformal invariant based
on a higher-derivative action for the linear multiplet. This invariant, which
includes gravitational curvature-squared terms, can be defined both coupled to
the standard-Weyl or dilaton-Weyl multiplet for conformal supergravity. In the
first case, we show how the addition of this invariant to the supersymmetric
Einstein-Hilbert term leads to a dynamically generated cosmological constant
and non-supersymmetric (A)dS$_6$ solutions. In the dilaton-Weyl multiplet, the
new off-shell invariant includes Ricci and scalar curvature-squared terms and
possesses a nontrivial dependence on the dilaton field.
