  Bismuth ferrite (BiFeO$_3$) is one of the very few known single-phase
multiferroic materials. While the bulk compound is rhombohedral (R), the
discovery of an epitaxial strain-induced structural transition into a so-called
'super tetragonal-phase' (T-phase) in this material incited a flurry of
research activity focused on gaining an understanding of this phase transition
and its possible functionalities. This metastable phase of BiFeO$_3$ is also
multiferroic, with giant ferroelectric polarization and coexisting
antiferromagnetic order, but above all it is the strain relaxation-induced
phase mixtures and their outstanding piezoelectric and magnetoelectric
responses which continue to intrigue and motivate the physicist and materials
scientist communities. Here, we review the research into the T-phase and
mixed-phase BiFeO$_3$ system. We begin with a brief summary of the history of
the T-phase and an analysis of the structure of the various phases reported in
the literature. We then address important questions regarding the symmetry and
octahedral rotation patterns and the (as yet underexplored) important role of
chemistry in the formation of the metastable T-phase. We follow by describing
the phase transitions in this material, and how these may hold promise for
large magnetoelectric responses. Finally we point out some experimental
challenges inherent to the study of such a system, and potential pathways for
how they may be overcome. It is our intention with this work to highlight
important issues that, in our opinion, should be carefully considered by the
community in order to use this fascinating materials system for a new paradigm
of functionality.
