A quantum observable which received renewed attention recently is entanglement entropy. It's application ranges over several fields in physics, from condensed matter physics to general relativity. In this dissertation we study entanglement entropy for quantum field theories in the presence of defects and singularities.

We study entanglement entropy using the framework of AdS/CFT correspondence. We focus on entangling surfaces across ball-shaped regions for systems outside their ground state. Quantum field theories in the presence of defects are considered first. These are the six-dimensional $(2,0)$ theory in the presence of Wilson surfaces and the four-dimensional $\cN = 4$ super-Yang-Mills theory in the presence of surface defects of the disordered type. Their holographic entanglement entropy is calculated applying the Ryu-Takayanagi prescripstion on their holographic duals, which are eleven-dimensional supergravity (M-theory) solutions for the former and ten-dimensional type IIB supergravity solutions for the latter. Other holographic observables are computed as well: the holographic stress tensor and the expectation value of the defect (operator). For the disordered defects, an alternative expression for the additional entanglement entropy due to the defect (in terms of expectation values) is derived, adapting the method of Lewkowycz and Maldacena for Wilson loops. The two entanglement entropies agree up to an additional term, the origin of which may be attributed to the conformal anomaly of even dimensional defects as we discuss.

The holographic entanglement and free energy is computed for five-dimensional super conformal field theories, starting from their holographic supergravity duals. Although the supergravity solutions possess singularities, these do not obstruct our calculations. The expected relation between the two observables is verified. This supports the supergravity solutions as holographic duals and gives the first quantitative results for five-dimensional superconformal field theories.