  We obtain an asymptotic formula for the average value of the operator product
expansion coefficients of any unitary, compact two dimensional CFT with $c>1$.
This formula is valid when one or more of the operators has large dimension or
-- in the presence of a twist gap -- has large spin. Our formula is universal
in the sense that it depends only on the central charge and not on any other
details of the theory. This result unifies all previous asymptotic formulas for
CFT$_2$ structure constants, including those derived from crossing symmetry of
four point functions, modular covariance of torus correlation functions, and
higher genus modular invariance. We determine this formula at finite central
charge by deriving crossing kernels for higher genus crossing equations, which
give analytic control over the structure constants even in the absence of exact
knowledge of the conformal blocks. The higher genus modular kernels are
obtained by sewing together the elementary kernels for four-point crossing and
modular transforms of torus one-point functions. Our asymptotic formula is
related to the DOZZ formula for the structure constants of Liouville theory,
and makes precise the sense in which Liouville theory governs the universal
dynamics of heavy operators in any CFT. The large central charge limit provides
a link with 3D gravity, where the averaging over heavy states corresponds to a
coarse-graining over black hole microstates in holographic theories. Our
formula also provides an improved understanding of the Eigenstate
Thermalization Hypothesis (ETH) in CFT$_2$, and suggests that ETH can be
generalized to other kinematic regimes in two dimensional CFTs.
