This talk will give a general overview of one of the most famous open problems for elliptic curves and one of the Millennium problems, the Birch and Swinnerton-Dyer conjecture. The conjectures states that the rank of the group of rational points coincides with the order of vanishing of the complex analytic L-function associated to the curve at s=1. After describing the group structure on the rational points on an elliptic curve, I will give a historical account on how exactly the conjecture was formulated in the 1960s by Brian Birch and Peter Swinnerton-Dyer based on computational experiments performed on the EDSAC computer at Cambridge University and will give several computational examples providing evidence for the conjecture. I will then briefly describe some of the current methods used to obtain results towards the conjecture. If time permits, I may discuss a recent result and a work in progress on the conjectural formula for elliptic curves over imaginary quadratic fields.