  We prove a long-standing conjecture on random-cluster models, namely that the
critical point for such models with parameter $q\geq1$ on the square lattice is
equal to the self-dual point $p_{sd}(q) = \sqrt q /(1+\sqrt q)$. This gives a
proof that the critical temperature of the $q$-state Potts model is equal to
$\log (1+\sqrt q)$ for all $q\geq 2$. We further prove that the transition is
sharp, meaning that there is exponential decay of correlations in the
sub-critical phase. The techniques of this paper are rigorous and valid for all
$q\geq 1$, in contrast to earlier methods valid only for certain given $q$. The
proof extends to the triangular and the hexagonal lattices as well.
