I will quickly review some basic facts of foliation theory followed by examples/constructions with naturally arising transverse geometric structures. Then I will introduce the holonomy groupoid and its reduced etale model as the framework to study such structures. I will emphasize the case for a transverse Riemannian structure and classical results from Molino's theory. As an application of these results, I will try to show the proof of the Hodge decomposition theorem for the appropriate transverse de-Rham cohomology for transversally Riemmanin foliations.