A mathematical model is called sparse if it is a combination of only a few non-zero terms. The aim of sparse interpolation is to determine both the support of the sparse linear combination and the coefficients in the representation, from a small or minimal amount of data samples. This talk centers around multi-exponential models: A common underlying problem statement in many applications is that of determining the number of exponential components, and for each component the value of the frequency, damping factor, amplitude and phase. It occurs, for instance, transient detection, motor fault diagnosis, electrophysiology, magnetic resonance spectroscopy, vibration analysis, fluorescence lifetime imaging, music signal processing, direction of arrival estimation in wireless communication systems, and so on. In this talk, we describe sparse interpolation. We focus on the connections between sparse interpolation, structured matrices, generalized eigenvalue computation, exponential analysis, and rational approximation. In the past few years, insight gained from the computer algebra community combined with methods developed by the numerical analysis community, has led to significant progress in several very practical and real-life signal processing applications. We make use of tools such as the singular value decomposition and various convergence results for Pade approximants to regularize an otherwise inverse problem. Classical resolution limitations in signal processing with respect to frequency and decay rates, are overcome. The connection with tensor decomposition leads to new possibilities to exploit sparsity in analyzing tensor-structured datasets.