Multi-exponential analysis might sound remote, but it touches our daily lives in many surprising ways, even if most people are unaware of how important it is. For example, a substantial amount of effort in signal processing and time series analysis is essentially dedicated to the analysis of multi-exponential functions. Multi- exponential analysis is also fundamental to several research fields and application domains that have been the subject of this Dagstuhl seminar: remote sensing, antenna design, digital imaging, all impacting some major societal or industrial challenges such as energy, transportation, space research, health and telecommunications. This Seminar connected stakeholders from seemingly separately developed fields: computational harmonic analysis, numerical linear algebra, computer algebra, nonlinear approximation theory, digital signal processing and their applications, in one and more variables.