  We compute the most general leading-order correction to Kerr solution when
the Einstein-Hilbert action is supplemented with higher-derivative terms,
including the possibility of dynamical couplings controlled by scalars. The
model we present depends on five parameters and it contains, as particular
cases, Einstein-dilaton-Gauss-Bonnet gravity, dynamical Chern-Simons gravity
and the effective action coming from Heterotic Superstring theory. By solving
the corrected field equations, we find the modified Kerr metric that describes
rotating black holes in these theories. We express the solution as a series in
the spin parameter $\chi$, and we show that including enough terms in the
expansion we are able to describe black holes with large spin. For the
computations in the text we use an expansion up to order $\chi^{14}$, which is
accurate for $\chi<0.7$, but we provide as well a Mathematica notebook that
computes the solution at any given order. We study several properties of the
corrected black holes, such as geometry of the horizon, ergosphere, light rings
and scalar hair. Some of the corrections violate parity, and we highlight in
those cases plots of horizons and ergospheres without $\mathbb{Z}_{2}$
symmetry.
