This talk is about a long-term project to verify the full Birch and Swinnerton-Dyer conjecture for every elliptic curve in Cremona's book, except the $18$ optimal curves of rank $2$. The methods involve Kolyvagin's Euler system of Heegner points, computation, and whatever else one can use. I'll begin with a discussion of the Birch and Swinnerton-Dyer conjecture for elliptic curves, and methods for computing the true order of Shafarevich-Tate groups, along with frustrating difficulties that have yet to be overcome.