The Hodge index theorem of Faltings and Hriljac asserts that the Neron-Tate height pairing on a projective curve over a number field is equal to certain intersection pairing in the setting of Arakelov geometry.In the talk,I will present an extension of the result to adelic line bundles on higher dimensional varieties over finitely generated fields.Then we will talk about its relation to the non-archimedean Calabi-Yau theorem and the its application to algebraic dynamics.This is a joint work with Shou-Wu Zhang.