We establish local well-posedness results for the Initial Value Problem associated to the Schrödinger-Debye system in dimensions
N
=
2
,
3
N=2, 3
for data in
H
s
×
H
ℓ
H^s\times H^{\ell }
, with
s
s
and
ℓ
\ell
satisfying
max
{
0
,
s
−
1
}
≤
ℓ
≤
min
{
2
s
,
s
+
1
}
\max \{0, s-1\} \le \ell \le \min \{2s, s+1\}
. In particular, these include the energy space
H
1
×
L
2
H^1\times L^2
. Our results improve the previous ones obtained by B. Bidégaray, and by A. J. Corcho and F. Linares. Moreover, in the critical case (
N
=
2
N=2
) and for initial data in
H
1
×
L
2
H^1\times L^2
, we prove that solutions exist for all times, thus providing a negative answer to the open problem mentioned by G. Fibich and G. C. Papanicolau concerning the formation of singularities for these solutions.