In this paper, we investigate the Variational Principle and develop 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 RW-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 RW-expansive. We also show a Ruelle--Perron--Frobenius Theorem for a family of hyperbolic holomorphic correspondences of the form $\textbf{f}_c (z)= z^{q/p}+c$.
