The design and performance of liquid metal batteries (LMBs), a new technology for grid-scale energy storage, depend on fluid mechanics because the battery electrodes and electrolytes are entirely liquid. Here, we review prior and current research on the fluid mechanics of LMBs, pointing out opportunities for future studies. Because the technology in its present form is just a few years old, only a small number of publications have so far considered LMBs specifically. We hope to encourage collaboration and conversation by referencing as many of those publications as possible here. Much can also be learned by linking to extensive prior literature considering phenomena observed or expected in LMBs, including thermal convection, magnetoconvection, Marangoni flow, interface instabilities, the Tayler instability, and electro-vortex flow. We focus on phenomena, materials, length scales, and current densities relevant to the LMB designs currently being commercialized. We try to point out breakthroughs that could lead to design improvements or make new mechanisms important.