**MSC:**- 14J25 Special surfaces, {For Hilbert modular surfaces, See 14G35}
- 14J10 Families, moduli, classification: algebraic theory

In this paper we provide a first step towards the

classification of the numerical Godeaux surfaces

in the still unknown open cases where $Tors(S) = 0$, or

$Tors(S) = \Z/2\Z$.

Our method works in both cases, but in this paper, after some results

which we establish in a greater generality, we mostly restrict ourselves to the

case where the Torsion group is zero.

The bicanonical system yields, on a

suitable blow up $\tilde{S}$ of the minimal model $S$, a fibration

$f:\tilde{S} \rightarrow {\mathbb P}^1$ in curves of genus $2\leq g \leq 4$,

and the invariants of this fibration determine the equations of the

image of $S$ under a map which, in the general case, is the

product of the tricanonical and of the bicanonical map.

This allows us to subdivide our surfaces into

four classes, according to the behaviour of the

bicanonical system. For each of these classes we have a complete

description but the existence questions are not yet solved.

\end{abstract}