  By focusing on the family $E:y^2=x^3+a$, we present strategies for
determining the structure of the torsion subgroup of the Mordell-Weil group of
an elliptic curve, $E(K)$, over quadratic field $K$. Generalizations of the
Nagell-Lutz theorem and Mazur's theorem to curves defined over quadratic fields
allows us to determine the full torsion subgroup of $E(K)$ as one of at most
three possibilities when $a$ is a square.
