Stated more than 20 years ago, the Heil, Ramanathan, and Topiwala (HRT) conjecture states "Each finite Gabor system generated by a nonzero square integrable function is a set of linearly independent functions." A Gabor system is a collection of time-frequency shifts of a fixed non zero measurable function. Despite the simplicity of its statement, the conjecture is still open for the general case. Based on a paper by J. Benedetto and the presenter, this talk presents results on HRT for finite Gabor systems generated by certain classes of elementary functions. For example, the paper proves HRT for an arbitrary Gabor system generated by exp(-|x|) or by any nonzero square integrable rational function.