  Epitaxial engineering of solid-state heterointerfaces is a leading avenue to
realizing enhanced or novel electronic states of matter. As a recent example,
bulk FeSe is an unconventional superconductor with a modest transition
temperature ($T_c$) of 9 K. When a single atomic layer of FeSe is grown on
SrTiO$_3$, however, its $T_c$ can skyrocket by an order of magnitude to 65 K or
109 K. Since this discovery in 2012, efforts to reproduce, understand, and
extend these findings continue to draw both excitement and scrutiny. In this
review, we first present a critical survey of experimental measurements
performed using a wide range of techniques. We then turn to the open question
of microscopic mechanisms of superconductivity. We examine contrasting
indications for both phononic (conventional) and magnetic/orbital
(unconventional) means of electron pairing, and speculations about whether they
could work cooperatively to boost $T_c$ in a monolayer of FeSe.
