Droplets of simple liquids, like oil and water, are essential engineering fluids. Classically, we use droplets to deliver material, transport heat, and control chemical reactions. Thanks to microfluidic technology, a wealth of new applications has emerged in recent years. Evolution, however, beat us to it. For more than a billion years, living cells have been producing microscopic droplets in the cytoplasm. These droplets have very different chemical compositions and mechanical properties than the simple liquids we encounter daily. They are thought to play an essential role in organizing chemical reactions in the cytoplasm. I will describe in vitro experiments demonstrating the remarkable efficiency of these droplets in localizing chemical reactions. When driven far from equilibrium, chemical reactions create stunning fluid motion, characterized by persistent cell-like motility and directed motion along chemical gradients. Further, I'll show how these droplets can also play a mechanical role within the cell. I will focus on one of these liquid phases, whose droplets are called stress granules. Cohesive forces holding stress granules together naturally lead to adhesive forces with other components of the cell. I'll show how how adhesion between stress granules and microtubules can reorganize the cytoskeleton.