  In this article we study superconductor-insulator transitions within the
general framework of an attractive Hubbard model. This is a well-defined model
of s-wave superconductivity which permits different tuning parameters (disorder
and field). Furthermore, it allows a comparison of various analytical and
computational approaches in order to gain a complete understanding of the
various effects of amplitude and phase fluctuations. We present a systematic
pedagogical approach, aiming to equip the lay reader with enough apparatus to
be able to understand the numerical calculations, reproduce some of the simpler
results, and be able to tackle future problems related to inhomogeneous phases.
We go into considerable detail on mean-field theory (MFT) and the Bogoliubov-de
Gennes (BdG) approach, as these are a first line of attack which can capture
much of the physics, but we also outline cases where this fails to capture
phase fluctuations and more sophisticated Quantum Monte Carlo (QMC)
calculations are necessary. We discuss the behavior of many observables,
including densities of states, superfluid stiffness, and dynamical
conductivity, for the disorder-tuned superconductor-insulator transition. We
also discuss SITs tuned by parallel magnetic field, which are quite different
due to pairbreaking.
