  We present the results of our search for the orders of Tate-Shafarevich
groups for the quadratic twists of elliptic curves. We formulate a general
conjecture, giving for a fixed elliptic curve $E$ over $\Bbb Q$ and positive
integer $k$, an asymptotic formula for the number of quadratic twists $E_d$,
$d$ positive square-free integers less than $X$, with finite group $E_d(\Bbb
Q)$ and $|\Sha(E_d(\Bbb Q))| = k^2$. This paper continues the authors previous
investigations concerning orders of Tate-Shafarevich groups in quadratic twists
of the curve $X_0(49)$. In section 8 we exhibit $88$ examples of rank zero
elliptic curves with $|\Sha(E)| > 63408^2$, which was the largest previously
known value for any explicit curve. Our record is an elliptic curve $E$ with
$|\Sha(E)| = 1029212^2$.
