Under non-equilibrium conditions, liquid droplets coupled to their environment by sustained flows of matter and/or energy can become "active" systems capable of various life-like functions. When "fueled" by even simple chemical reactions, such droplets can become tactic and can perform "intelligent" tasks such as maze solving. With more complex chemistries, droplets can support basic forms of metabolism, grow, self-replicate, and exhibit evolutionary changes akin to biological cells. There are also first exciting examples of active droplets connected into larger, tissue-like systems supporting droplet-to-droplet communication, and giving rise to collective material properties. As practical applications of droplets also begin to appear (e.g., in single-cell diagnostics, new methods of electricity generation, optofluidics, or sensors), it appears timely to review and systematize progress in this highly interdisciplinary area of chemical research, and also think about the avenues (and the roadblocks) for future work.