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 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.