  In this survey, we review the literature on inverse problems in topological
persistence theory. The first half of the survey is concerned with the question
of surjectivity, i.e. the existence of right inverses, and the second half
focuses on injectivity, i.e. left inverses. Throughout, we highlight the tools
and theorems that underlie these advances, and direct the reader's attention to
open problems, both theoretical and applied.
