  In this thesis, we examine in detail the notion of black hole entropy in
Quantum Field Theories, with a specific focus on supersymmetric black holes and
the perturbative and non-perturbative quantum corrections to the classical
area-law of Bekenstein-Hawking. To examine such corrections, we employ the
formalism of Sen's Quantum Entropy Function where the complete quantum entropy
of a supersymmetric black hole in theories of supergravity is defined as a
path-integral in the near-horizon region of the black hole. Evaluation of this
path-integral can then be conducted exactly using localization computation
techniques. Due to the exactness of the localization argument, the results
obtained in this manner are therefore formally expected to re-sum all
perturbative and non-perturbative corrections to the classical area-law, and
thus connect to string-theoretic predictions. We investigate such connections
in detail for specific supersymmetric black holes in the hopes of strengthening
a Boltzmann-type interpretation of their thermodynamic entropy as arising from
the degeneracies of the microscopic gravitational constituents (the D-branes).
We find that this picture holds very precisely for two types of black holes
preserving four real supercharges in both four-dimensional $\mathcal{N}=8$ and
$\mathcal{N}=4$ string theories and supergravities. From a broader point of
view, such results can be interpreted as providing important examples where
supergravity theories encode the complete low-energy dynamics of string
theories and provide a consistent effective picture. Some interesting
connections to the mathematical theory of modular forms and mock modular forms
are also exhibited.
