  A noncommutative Feynman graph is a ribbon graph and can be drawn on a genus
$g$ 2-surface with a boundary. We formulate a general convergence theorem for
the noncommutative Feynman graphs in topological terms and prove it for some
classes of diagrams in the scalar field theories. We propose a noncommutative
analog of Bogoliubov-Parasiuk's recursive subtraction formula and show that the
subtracted graphs from a class $\Omega_d$ satisfy the conditions of the
convergence theorem. For a generic scalar noncommutative quantum field theory
on $\re^d$, the class $\Omega_d$ is smaller than the class of all diagrams in
the theory. This leaves open the question of perturbative renormalizability of
noncommutative field theories. We comment on how the supersymmetry can improve
the situation and suggest that a noncommutative analog of Wess-Zumino model is
renormalizable.
