This introduction to inverse problems for acoustic or elastic waves does not aim to be an exhaustive presentation of their theoretical and numerical difficulties, nor of their applications in fields of interest—including those of evaluation and non-destructive testing (NDT) of artificial structures, or the characterization of the human body, two topics among those discussed in detail in other chapters of this book. We only wish to highlight general elements that can be found in a relevant way in many of the inverse problems we can meet in practice. Perhaps more importantly, we also wish to give to the interested reader some guidelines for further research, based on a list of publications from the recent international literature, in addition to some examples from different works to which the authors have been contributing in acoustic and elastic topological identification, an investigation which we believe to be useful for our purpose here.