We begin by discussing the circumstances of capillary surfaces in regions with a corner, both concave and convex. These circumstances led to the (since proved) Concus-Finn Conjecture, which gives the requirements for a continuous solution to the capillary problem. We include the requirements for the existence of radial limits and fans of radial limits in these corner regions. We outline the method for which conformal mapping of the Gauss map of these surfaces can be computed to allow the analytic extension of many related theorems. Finally, we introduce a broadening of the existing research by taking an example done with Dirichlet conditions in 1996 by Kirk Lancaster and David Siegel and recreating the example with contact angle data instead. With the use of contact angle data we find that where in the original example γ could not be bounded away from zero or π, we were able to find a capillary surface such that γ is bounded away from 0 and π.