  Hexagonal $AB$O$_3$ oxides ($A$, $B$ = cation) are a rich materials class for
realizing novel quantum phenomena. Their hexagonal symmetry, oxygen trigonal
bipyramid coordination and quasi-two dimensional layering give rise to
properties distinct from those of the cubic $AB$O$_3$ perovskites. As bulk
materials, most of the focus in this materials class has been on the rare earth
manganites, $R$MnO$_3$ ($R$ = rare earth); these materials display coupled
ferroelectricity and antiferromagnetic order. In this review, we focus on the
thin film manifestations of the hexagonal $AB$O$_3$ oxides. We cover the
stability of the hexagonal oxides and substrates which can be used to template
the hexagonal structure. We show how the thin film geometry not only allows for
further tuning of the bulk-stable manganites but also the realization of
metastable hexagonal oxides such as the $R$FeO$_3$ that combine
ferroelectricity with weak ferromagnetic order. The thin film geometry is a
promising platform to stabilize additional metastable hexagonal oxides to
search for predicted high-temperature superconductivity and topological phases
in this materials class.

Materials Science