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