**MSC:**- 14P25 Topology of real algebraic varieties
- 14J15 Moduli, classification: analytic theory, See also {32G13,

real algebraic surfaces of Kodaira dimension zero and we make a step

towards the Enriques classification of real algebraic surfaces, by

describing in detail the structure of the moduli space of real

hyperelliptic surfaces.

Moreover, we point out the relevance in real geometry of the notion

of the orbifold fundamental group of a real variety, and we discuss

related questions on real varieties $(X, \sigma)$ whose underlying

complex manifold $X$ is a $K ( \pi, 1)$.

Our first result is that if $(S, \sigma)$ is a real hyperelliptic

surface, then the differentiable type of the pair $(S,

\sigma)$ is completely determined by the orbifold fundamental group

exact sequence. \\ This result allows us to determine all the

possible topological types of

$(S, \sigma)$, and to prove that they are exactly 78.

It follows also as a corollary that there are exactly eleven cases

for the topological type of the real part of S.

Finally, we show that once we fix the topological type of $(S,

\sigma)$ corresponding to a real hyperelliptic surface, the

corresponding moduli space is irreducible (and connected).

We also give, through a series of tables, explicit analytic

representations of the 78 components of the moduli space.