The design and performance of liquid metal batteries, 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 liquid metal batteries, 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 liquid metal batteries 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 liquid metal batteries, 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 liquid metal battery designs currently being commercialized. We try to point out breakthroughs that could lead to design improvements or make new mechanisms important.The story of fluid mechanics research in liquid metal batteries (LMBs) begins with one very important application: grid-scale storage. Electrical grids have almost no energy storage capacity, and adding storage will make them more robust and more resilient even as they incorporate increasing amounts of intermittent and unpredictable wind and solar generation. Liquid metal batteries have unique advantages as a grid-scale storage technology, but their uniqueness also means that designers must consider chemical and physical mechanisms -including fluid mechanisms -that are relevant to few other battery technologies, and in many cases not yet well-understood. We will review the fluid mechanics of liquid metal batteries, focusing on studies undertaken with that technology in mind, and also drawing extensively from prior work considering similar mechanisms in other contexts. In the interest of promoting dialogue across this new field, we have endeavored to include the work of many different researchers, though inevitably some will have eluded our search, and we ask for the reader's sympathy for regrettable omissions. Our story will be guided by technological application, focusing on mechanisms most relevant to liquid metal batteries as built for grid-scale storage. We will consider electrochemistry and theoretical fluid mechanics only briefly because excellent reviews of both topics are already available in the literature. In §1 below we provide an overview and brief introduction to liquid metal batteries, motivated by the present state of worldwide electrical grids, including the various types of liquid metal batteries that have been developed. We consider the history of liquid metal batteries in more detail in §2, connecting to the thermally regenerative electrochemical cells developed in the middle of the twentieth century. Continuing, we consider the fluid mechanisms that are most relevant to ...