BiFeO3 is perhaps the only material that is both magnetic and a strong ferroelectric at room temperature. As a result, it has had an impact on the field of multiferroics that is comparable to that of yttrium barium copper oxide (YBCO) on superconductors, with hundreds of publications devoted to it in the past few years. In this Review, we try to summarize both the basic physics and unresolved aspects of BiFeO3 (which are still being discovered with several new phase transitions reported in the past few months) and device applications, which center on spintronics and memory devices that can be addressed both electrically and magnetically. Despite the attention devoted to multiferroic BiFeO3 in recent years, many fundamental aspects, such as the phase diagram, are still poorly understood. The phase diagram in the figure, for example, includes several confirmed phase transitions, but does not account for other observed anomalies both above and below room temperature. These unresolved issues, as well as progress made towards applications, are discussed in this Review.