  We investigate the projective normality of smooth, linearly normal surfaces
of degree 9. All non projectively normal surfaces which are not scrolls over a
curve are classified. Results on the projective normality of surface scrolls
are also given. One of the reasons that brought us to look at this question is
our desire to find examples for a long standing problem in adjunction theory.
Andreatta followed by a generalization by Ein and Lazarsfeld posed the problem
of classifying smooth n-dimensional varieties (X,L) polarized with a very ample
line bundle L, such that the adjoint linear system |H| = |K + (n-1)L| gives an
embedding which is not projectively normal. After a detailed check of the non
projectively normal surfaces found in this work no examples were found except
possibly a blow up of an elliptic P^1-bundle whose existence is uncertain.
