As has been known since the time of Gromov's Nonsqueezing Theorem, questions about symplectic embeddings lie at the heart of symplectic geometry. In the past few years we have gained significant new insight into the question of when there is a symplectic embedding of one basic geometric shape (such as a ball or ellipsoid) into another (such as an ellipsoid or torus). This talk will describe some of this progress, with particular emphasis on results in dimension four.