Renormalized Volume and Area

In this dissertation we look at some problems on renormalized volume and area. We develop a Chern-Gauss-Bonnet formula for the renormalized area of a hypersurface within a 5-dimensional Poincaré-Einstein manifold. This is followed by an investigation into renormalized area in the Singular Yamabe setting. Lastly, we look at the renormalized volume of a minimally bounded region within a 4-dimensional Poincaré-Einstein manifold.