  We define two concordance invariants of knots using framed instanton
homology. These invariants $\nu^\sharp$ and $\tau^\sharp$ provide bounds on
slice genus and maximum self-linking number, and the latter is a concordance
homomorphism which agrees in all known cases with the $\tau$ invariant in
Heegaard Floer homology. We use $\nu^\sharp$ and $\tau^\sharp$ to compute the
framed instanton homology of all nonzero rational Dehn surgeries on: 20 of the
35 nontrivial prime knots through 8 crossings, infinite families of twist and
pretzel knots, and instanton L-space knots; and of 19 of the first 20 closed
hyperbolic manifolds in the Hodgson--Weeks census. In another application, we
determine when the cable of a knot is an instanton L-space knot. Finally, we
discuss applications to the spectral sequence from odd Khovanov homology to the
framed instanton homology of branched double covers, and to the behaviors of
$\tau^\sharp$ and $\tau$ under genus-2 mutation.
