The elegant theory developed by Forster to describe the rate of fluorescence resonance energy transfer between a donor and an acceptor has played a key role in understanding the structure and dynamics of polymers, biopolymers (proteins, nucleic acids), and self-assemblies (photosystems, micellar systems). Forster theory assumes the transition charge densities of donor and acceptor molecules are point dipoles and hence predicts a 1/R-6 dependence of energy transfer rate on center-to-center separation distance, R. In addition, a preaveraging over the orientations of the two dipoles is usually performed. The present review examines the validity of these assumptions in following different donor-acceptor (D-A) systems: (i) dye molecules attached to a flexible polymer chain in solution, (ii) extended conjugated dye molecules in quenched conformation, (iii) dye and a spherical metal nanoparticle of different sizes, (iv) two spherical metal nanoparticles, and (v) two prolate shaped metal nanoparticles at different relative orientations. In the case of dye molecules attached to a flexible polymer chain, we discuss the recent theoretical and computer simulation studies of energy transfer dynamics. It includes an analysis of Wilemski-Fixman (WF) theory of a bimolecular reaction in solution, applied to the excitation energy transfer between two ends of the polymer. We briefly describe the limitation of the WF theory and its generalizations that lead to a better agreement between the theory and the simulation results. The orientational dynamics of dye molecules is found to significantly influence the rate of excitation energy transfer, and may play a "hidden role" in influencing the observed distance dependence. For extended conjugated D-A systems and those involving nonspherical metal nanoparticles, even at intermediate separations, a significant deviation from 1/R-6-type distance dependence of the energy transfer rate is found. Surprisingly, however, this distance dependence is robust for D-A systems involving spherical metal nanoparticles. For both spherical and nonspherical metal nanoparticles (MNps), the functional dependence of rate on the surface-to-surface separation distance (d) is quite different, at small to intermediate distances (compared to the size of the MNps). The rate calculations of excitation energy transfer between extended conjugated dye molecules reveal that optically dark states can significantly contribute toward enhancing the energy transfer rate. It is further found that the rate of energy transfer between nonspherical metal nanoparticles exhibits an interesting orientation dependence not anticipated in Forster's approach.