  We derive rigorous bounds on corrections to Einstein gravity using unitarity
and analyticity of graviton scattering amplitudes. In $D\geq 4$ spacetime
dimensions, these consistency conditions mandate positive coefficients for
certain quartic curvature operators. We systematically enumerate all such
positivity bounds in $D=4$ and $D=5$ before extending to $D\geq 6$. Afterwards,
we derive positivity bounds for supersymmetric operators and verify that all of
our constraints are satisfied by weakly-coupled string theories. Among
quadratic curvature operators, we find that the Gauss-Bonnet term in $D\geq 5$
is inconsistent unless new degrees of freedom enter at the natural cutoff scale
defined by the effective theory. Our bounds apply to perturbative ultraviolet
completions of gravity.
