  In this work, we investigate the Variational Principle and develop a
thermodynamic formalism for correspondences. We define the measure-theoretic
entropy for transition probability kernels and topological pressure for
correspondences. Based on these two notions, we establish the following
results:
  The Variational Principle holds and equilibrium states exist for continuous
potential functions, provided that the correspondence satisfies some expansion
property called forward expansiveness. If, in addition, the correspondence
satisfies the specification property and the potential function is Bowen
summable, then the equilibrium state is unique. On the other hand, for a
distance-expanding, open, strongly transitive correspondence and a H\"{o}lder
continuous potential function, there exists a unique equilibrium state, and the
backward orbits are equidistributed. Furthermore, we investigate the
Variational Principle for general correspondences.
  In conformal dynamics, we establish the Variational Principle for the
Lee--Lyubich--Markorov--Mazor--Mukherjee anti-holomorphic correspondences,
which are matings of some anti-holomorphic rational maps with anti-Hecke groups
and not forward expansive. We also show a Ruelle--Perron--Frobenius Theorem for
a family of hyperbolic holomorphic correspondences of the form
$\boldsymbol{f}_c (z)= z^{q/p}+c$.
