In this talk, we will examine the effect that quadratic base change of an elliptic curve E over a number field K has on its torsion subgroup. Over K=Q, it has been extensively studied. As a step toward number fields greater than Q, we will start with elliptic curves defined over a fixed quadratic field. The proofs require the use of several techniques ranging from the Galois action to the study of quadratic points on certain modular curves.