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Compressive sensing (CS) is a novel sampling paradigm that samples signals in a much more efficient way than the established Nyquist sampling theorem. CS has recently gained a lot of attention due to its exploitation of signal sparsity. Sparsity, an inherent characteristic of many natural signals, enables the signal to be stored in few samples and subsequently be recovered accurately, courtesy of CS. This article gives a brief background on the origins of this idea, reviews the basic mathematical foundation of the theory and then goes on to highlight different areas of its application with a major emphasis on communications and network domain. Finally, the survey concludes by identifying new areas of research where CS could be beneficial.