Deviations from the usual $R^{-6}$ dependence of the rate of fluorescence resonance energy transfer (FRET) on the distance between the donor and the acceptor have been a common scenario in the recent times. In this paper, we present a critical analysis of the distance dependence of FRET, and try to illustrate the non-$R^{-6}$ type behaviour of the rate for the case of transfer from a localized electronic excitation on the donor, a dye molecule to three different energy acceptors with delocalized electronic excitations namely, graphene, a two-dimensional semiconducting sheet and the case of such a semiconducting sheet rolled to obtain a nanotube. We use simple analytic models to understand the distance dependence in each case.