  We study the Cauchy problem of the Schr\"odinger-Korteweg-de Vries system.
First, we establish the local well-posedness results, which improve the results
of Corcho, Linares (2007). Moreover, we obtain some ill-posedness results,
which show that they are sharp in some well-posedness thresholds. Particularly,
we obtain the local well-posedness for the initial data in
$H^{-{3/16}+}(\R)\times H^{-{3/4}+}(\R)$ in the resonant case, it is almost the
optimal except the endpoint. At last we establish the global well-posedness
results in $H^s(\R)\times H^s(\R)$ when $s>\dfrac{1}{2}$ no matter in the
resonant case or in the non-resonant case, which improve the results of Pecher
(2005).
