This paper begins with a review of some classical work of Landau, Slepian, Pollak and Widom concerning essentially time-and bandlimited signals and ends reviewing some recent work of Candès, Romberg and Tao that places specific but probabilistic limitations on essential time-and bandlimiting for finite signals and their discrete Fourier transforms. In between we outline some conceptual bridges from the continuous, singleband setting to the finite, multiband setting and pose a number of open problems whose solutions would solidify the connections outlined. These connections involve time-frequency localization of multiband signals and sampling theory for such signals.

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