Modern versions of exponential analysis solve the problem in two steps: first a Hankel structured generalized eigenvalue problem reveals the exponential parameters and next a Vandermonde linear system is solved for the linear coefficients in the exponential model. In the last decade several pressing open problems in exponential analysis were tackled. Among others, we mention the reliable computation of the model's sparsity or number of terms, the addition of robustness to algortihms for this numerically sensitive problem, a multidimensional version with similar low data usage, and superresolution allowing to separate clustered terms. The new results proved very valuable in several applications in signal processing, antenna engineering, localisation problems, texture analysis, and more.