Abstract: Elliptic curves are fundamental objects in modern number theory and one fruitful way to study them is through their Galois representations. These representations arise by considering the natural Galois action on the torsion points of the curve. For a typical elliptic curve defined over the rationals, a famous theorem of Serre says that the Galois action on the torsion points is "almost as large as possible". After going over some background, we will state a precise version of Serre's theorem and describe Mazur's "Program B". We will then discuss a recent approach to proving a more quantitative version of Serre's result.