  We provide a complete solution to the problem of extending arbitrary semistar
operations of an integral domain $D$ to semistar operations of the polynomial
ring $D[X]$. As an application, we show that one can reobtain the main results
of some previous papers concerning the problem in the special cases of stable
semistar operations of finite type or semistar operations defined by families
of overrings. Finally, we investigate the behavior of the polynomial extensions
of the most important and classical operations such as $d_D$, $v_D$, $t_D$,
$w_D$ and $b_D$ operations.
