  For an elliptic curve $E/\Q$, we determine the maximum number of twists
$E^d/\Q$ it can have such that $E^d(\Q)_{tors}\supsetneq E(\Q)[2]$. We use
these results to determine the number of distinct quadratic fields $K$ such
that $E(K)_{tors}\supsetneq E(\Q)_{tors}$. The answer depends on $E(\Q)_{tors}$
and we give the best possible bound for all the possible cases.
