  Consider a sequence of closed, orientable surfaces of fixed genus $g$ in a
Riemannian manifold $M$ with uniform upper bounds on mean curvature and area.
We show that on passing to a subsequence and choosing appropriate
parametrisations, the inclusion maps converge in $C^0$ to a map from a surface
of genus $g$ to $M$.
  We also show that, on passing to a further subsequence, the distance
functions corresponding to pullback metrics converge to a pseudo-metric of
fractal dimension two.
  As a corollary, we obtain a purely geometric result. Namely, we show that
bounds on the mean curvature, area and genus of a surface $F\subset M$ together
with bounds on the geometry of $M$ give an upper bound on the diameter of $F$.
  Our proof is modelled on Gromov's compactness theorem for $J$-holomorphic
curves.
