  Wannier functions have widespread utility in condensed matter physics and
beyond. Topological physics, on the other hand, has largely involved the
related notion of compactly-supported Wannier-type functions, which arise
naturally in flat bands. In this paper, we establish a connection between these
two notions, by finding the necessary and sufficient conditions under which
compact Wannier functions exist in one dimension. We present an exhaustive
construction of models with compact Wannier functions and show that the Wannier
functions are unique, and in general, distinct from the corresponding
maximally-localized Wannier functions.
