Full waveform inversion (FWI), despite its wide adoption as part of the seismic imaging workflow in the hydrocarbon exploration industry, relies on the solution of a nonlinear and ill-posed inverse problem. For this reason, successful applications of FWI still require non-negligible human intervention to design proper initial models and adequate hierarchy in the data ensuring the convergence towards a geologically meaningful minimum of the misfit function. For specific targets, the complexity of the wavefield can still prevent FWI to converge towards such a minimum. This is the motivation for more than two decades to propose alternative formulations to FWI, alleviating its ill-posedness nature. In this study, we attempt to propose a general review of these formulations, based on a unifying mathematical framework, which makes possible to highlight similarities and differences between the different approaches. A discussion summarizes this comparison, with a focus on the applicability of the proposed methods to the inversion of field data.