In a very influential paper on the theory of signal communication which appeared in 1946, Dennis Gabor who later won the Nobel prize for his seminal work in laser holography, made the following claim. Linear combinations of regularly spaced discrete time and frequency shifts of the Gaussian function form a stable reproducing system for the space of signals of finite energy. Gabor's idea was primarily motivated by the need to construct building blocks with optimal time-frequency concentration. Similar themes also arose naturally and independently in the context of quantum mechanics and electrical engineering. It also seems that Gabor thought that time-frequency shifts of the Gaussian form a basis for all signals of finite energy. It turned out that in a mathematically rigorous sense, Gabor's claim was false. Since, the space of finite energy signals is infinite-dimensional, spanning sets are not generally bases. Although time-frequency shifts of the Gaussian form a complete system as shown by von Neumann, in a sense which we shall make precise, they do not form a basis. The aim of this talk is to provide a clear and transparent proof of the reasons why Gabor was wrong. Connection will be made with current and relevant questions such as the HRT conjecture (open question.) If time allows, we will go over some of the challenges involved in settling the HRT conjecture. Students who are interested in being involved in undergraduate research are encouraged to attend the presentation.