For an elliptic curve E over the rationals, the Birch--Swinnerton-Dyer conjecture predicts that (a) the rank of the Mordell--Weil group E(Q) equals the order of vanishing at s=1 of the L-function L(E,s) and that (b) the leading non-zero Taylor series coefficient of L(E,s) around s=1 is given by a formula in terms of objects associated with E (such as the order of its Tate-Shafarevich group, etc). This talk will explain the strategy and ingredients in recent proofs of the p-part of (b) for most odd primes p when the curve E has analytic rank 0 or 1 (so (a) is also known). Speaker(s): Christopher Skinner (Princeton University)