Elliptic curves are fundamental objects in number theory and a 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 an elliptic curve defined over a number field, a famous theorem of Serre says that the Galois action on the torsion points is "almost as large as possible". After some review and motivation, we will state a precise version of Serre's theorem and speculate on what may true. We will give several examples and briefly mention modular curves.