In the theory of linear second order partial differential equations with nonnegative characteristic form, the study of H ̈ormander operators occupies a preeminent position, both because they arise in many different theoretical and applied areas, and because of the richness of the results obtained up to the present time. Several monographs and survey memoirs concerning H ̈ormander operators are today available in the literature. However, most of them only consider and illustrate particular, even though important, operators such as sub-Laplacians or heat-type operators. More- over, the valuable monograph by Olejnik and Radkevic, the only one concerning general linear second order PDEs with nonnegative characteristic form, dates back to many years ago and does not take into account the remarkable geometric structures underlying the relevant operators which today provide many important tools to develop the broad and profound theory of H ̈ormander’s PDEs.