During the past few years a considerably research effort has been devoted to the problem of identifying parsimonious models from experimental data. Since this problem is generically non-convex, these approaches typically rely on relaxations such as Group Lasso or nuclear norm minimization. However, while these approaches usually work well in practice, there is no guarantee that using these surrogates will lead to the simplest model explaining the experimental data. In addition, incorporating stability constraints into the formalism entails a substantial increase in the computational complexity. Alternatively stability and model order constraints can be handled directly using a moments based approach. However, presently this approach is limited to relatively small sized problems, due to its computational complexity. Motivated by these difficulties, recently a new approach has been proposed based on the idea of representing the response of an LTI system as a linear combination of suitably chosen objects (atoms) and the observation that minimizing the atomic norm leads to sparse representations. In this paper we cover the fundamentals of this new approach and show that it leads to a very efficient algorithm, that avoids the need for using regularization steps and automatically incorporates stability constraints. In addition, this approach can be extended to accommodate non-uniform sampling and (unknown) initial conditions. These results are illustrated with several examples, including identification of a very lightly damped structure from time and frequency domain measurements.