This dissertation investigates extensions of AdS/CFT and its applications to black holes and condensed matter. We begin by introducing the AdS/CFT correspondence and the various numerical methods we employ. We then apply these methods to construct a 'black funnel'. These solutions are dual to a Hartle-Hawking state of a strongly coupled field theory on a black hole background.

We then study the Lifshitz spacetime which is dual to a non-relativistic field theory with an anisotropic scaling symmetry. This spacetime has a curvature singularity due to diverging tidal forces. We examine the properties of a test string as it propagates in this background and find that the singularity causes the string to experience infinite mode production.

We explore the effects of a confinement/deconfinement transition. In the Lifshitz spacetime, this additional ingredient makes the zero temperature limit singularity-free. We also add a confinement/deconfinement transition to the holographic superconductor and produce its phase diagram. This system has a rich phase structure with a conducting phase, an insulating phase, a confining superconductor, and a deconfined superconductor. Surprisingly, it also allows for a situation where a superconductor can transition to an insulator by lowering the temperature.

We attempt to reproduce some known results in condensed matter from a holographic standpoint. We construct a holographic model of a Josephson junction and reproduce the familiar properties associated with it. We then study the effects of finite size on holographic superconductors and find a behaviour consistent with the known results of strongly coupled superconductors.