  Emergent electronic phenomena in iron-based superconductors have been at the
forefront of condensed matter physics for more than a decade. Much has been
learned about the origin and intertwined roles of ordered phases, including
nematicity, magnetism, and superconductivity, in this fascinating class of
materials. In recent years, focus has been centered on the peculiar and highly
unusual properties of FeSe and its close cousins. This family of materials has
attracted considerable attention due to the discovery of unexpected
superconducting gap structures, a wide range of superconducting critical
temperatures, and evidence for nontrivial band topology, including associated
spin-helical surface states and vortex-induced Majorana bound states. Here, we
review superconductivity in iron chalcogenide superconductors, including bulk
FeSe, doped bulk FeSe, FeTe$_{1-x}$Se$_x$, intercalated FeSe materials, and
monolayer FeSe and FeTe$_{1-x}$Se$_x$ on SrTiO$_3$. We focus on the
superconducting properties, including a survey of the relevant experimental
studies, and a discussion of the different proposed theoretical pairing
scenarios. In the last part of the paper, we review the growing recent evidence
for nontrivial topological effects in FeSe-related materials, focusing again on
interesting implications for superconductivity.
