  In this work I study the well-posedness of the Cauchy problem associated with
the coupled Schr\"odinger equations {with quadratic nonlinearities}, which
appears modeling problems in nonlinear optics. I obtain the local
well-posedness for data {in Sobolev spaces} with low regularity. {To obtain}
the local theory, I prove new bilinear estimates for the coupling terms of the
system in the continuous case. Concerning global results, in the continuous
case, I establish the global well-posedness in $H^s(\mathbb{R})\times
H^s(\mathbb{R})$, for some negatives indexes $s$. The proof of the global
result uses the \textbf{I}-method introduced by Colliander, Keel, Staffilani,
Takaoka and Tao.
