Suppose one wants to recover an unknown signal x in Rn from a given vector Ax=b in Rm of linear measurements of the signal x. If the number of measurements m is less than the degrees of freedom n of the signal, then the problem is underdetermined and the solution x is not unique. However, if we also know that x is sparse or compressible with respect to some basis, then it is a remarkable fact that (given some assumptions on the measurement matrix A) we can reconstruct x from the measurements b with high accuracy, and in some cases with perfect accuracy. Furthermore, the algorithm for performing the reconstruction is computationallyfeasible. This observation underlies the newly developing field of compressed sensing. In this talk we will discuss some of the mathematical foundations of this.