Neutrinos are among the most abundant particles in the universe, yet there is a surprising amount of information we still do not know about them. The discovery of neutrino masses and mixing over a decade ago has raised a large number of challenging questions about neutrinos and their connections to the world we live in. After briefly reviewing what we have learned about neutrinos so far, we will examine these open questions, explain why they are interesting, and discuss plans for answering them in future experiments.