  We propose new types of canonical metrics on K\"ahler manifolds, called
coupled K\"ahler-Einstein metrics, generalizing K\"ahler-Einstein metrics. We
prove existence and uniqueness results in the cases when the canonical bundle
is ample and when the manifold is K\"ahler-Einstein Fano. In the Fano case we
also prove that existence of coupled K\"ahler-Einstein metrics imply a certain
algebraic stability condition, generalizing $K$-polystability.
