In our talk we will give a panorama of José Burgos' contributions to various generalizations of the classical arithmetic intersection theory developed by Gillet and Soulé. It starts with the extension of Arakelov geometry allowing to incorporate logarithmically singular metrics with applications to Shimura varieties. Further generalizations include toric varieties as well as the most recent results about arithmetic intersections of arithmetic b-divisors with applications to mixed Shimura varieties including the theory of Siegel-Jacobi forms.[-]