  Making suitable generalizations of known results we prove some general facts
about Gaussian maps. The above are then used, in the second part of the
article, to give a set of conditions that insure the surjectivity of Gaussian
maps for curves on Enriques surfaces. To do this we also solve a problem of
independent interest: a tetragonal curve of genus g > 6 lying on an Enriques
surface and general in its linear system, cannot be, in its canonical
embedding, a quadric section of a surface of degree g - 1 in P^{g - 1}.
