  Inspired by a construction by Arnaud Beauville of a surface of general type
with $K^2 = 8, p_g =0$, the second author defined the Beauville surfaces as the
surfaces which are rigid, i.e., they have no nontrivial deformation, and admit
un unramified covering which is isomorphic to a product of curves of genus at
least 2.
  In this case the moduli space of surfaces homeomorphic to the given surface
consists either of a unique real point, or of a pair of complex conjugate
points corresponding to complex conjugate surfaces. It may also happen that a
Beauville surface is biholomorphic to its complex conjugate surface, neverless
it fails to admit a real structure.
  First aim of this note is to provide series of concrete examples of the
second situation, respectively of the third. Second aim is to introduce a wider
audience, especially group theorists, to the problem of classification of such
surfaces, especially with regard to the problem of existence of real structures
on them.
