  This is an expository article, based on a lecture course given at CRM
Barcelona in December 2009. The purpose of these notes is to prove, in a
reasonably self-contained way, that finiteness of the Tate-Shafarevich group
implies the parity conjecture for elliptic curves over number fields. Along the
way, we review local and global root numbers of elliptic curves and their
classification, and discuss some peculiar consequences of the parity
conjecture.
