It has been known since the time of Gromov that questions about symplectic embeddings lie at the heart of symplectic geometry. This talk will mostly be about some recent work with Schlenk in which we work out precisely when a four dimensional ellipsoid embeds symplectically in a ball. This problem turns out to have unexpected relations with the properties of continued fractions and of exceptional curves in blow ups of the complex projective plane. It is also related to questions of lattice packing of planar triangles.