I will discuss contributions to the following open problem:  which right-angled Artin groups (RAAGs) contain the fundamental group of a closed hyperbolic surface as a subgroup.  The first result in this direction, due to Droms, Servatius & Servatius is easy to state:  if the RAAG is defined by a graph which contains a full circuit of length greater than four, then there is a hyperbolic surface subgroup.  Recent work by Kim and by Crisp, Sageev & Sapir has demonstrated that the general problem is much more delicate.  I will outline the current status of the problem and illustrate some new examples.