The theory of Compressed Sensing, the emerging sampling paradigm 'that goes against the common wisdom', asserts that 'one can recover signals in Rn from far fewer samples or measurements, if the signal has a sparse representation in some orthonormal basis', from m = O(klogn), k<< n nonadaptive measurements . The accuracy of the recovered signal is 'as good as that attainable with direct knowledge of the k most important coefficients and its locations'. Moreover, a good approximation to those important coefficients is extracted from the measurements by solving a L1 minimization problem viz. Basis Pursuit. 'The nonadaptive measurements have the character of random linear combinations of the basis/frame elements'. The theory has implications which are far reaching and immediately leads to a number of applications in Data Compression,Channel Coding and Data Acquisition. 'The last of these applications suggest that CS could have an enormous impact in areas where conventional hardware design has significant limitations', leading to 'efficient and revolutionary methods of data acquisition and storage in future'. The paper reviews fundamental mathematical ideas pertaining to compressed sensing viz. sparsity, incoherence, reduced isometry property and basis pursuit, exemplified by the sparse recovery of a speech signal and convergence of the L1- minimization algorithm.