It is well known that Riemannian submersions are of interest in physics, owing to their applications in the Yang-Mills theory, Kaluza-Klein theory, supergravity and superstring theories. In this paper we give a survey of harmonic maps and Riemannian submersions between manifolds equipped with certain geometrical structures such as almost Hermitian structures, contact structures, f-structures and quaternionic structures. We also present some new results concerning holomorphic maps and semi-Riemannian...