Abstract: Central objects in the study of elliptic curves are the fields of definition of the points of order n. Relatively little is known about the way in which a fixed elliptic curve's division fields can intersect. In this talk, we lay the foundations for a systematic study of the entanglements of division fields of elliptic curves from a group theoretic perspective. In particular, we classify the way in which two prime-level division fields can intersect for infinitely many elliptic curves. This is joint work with Jackson S. Morrow.