  Implicit in the study of magnetic materials is the concept of spin
Hamiltonians, which emerge as the low-energy theories of correlation-driven
insulators. In order to predict and establish such Hamiltonians for real
materials, a variety of first principles $ab$-$initio$ methods have been
developed, based on density functional theory and wavefunction methodologies.
In this review, we provide a basic introduction to such methods and the
essential concepts of low-energy Hamiltonians, with a focus on their practical
capabilities and limitations. We further discuss our recent efforts toward
understanding a variety of complex magnetic systems that present unique
challenges from the perspective of $ab$-$initio$ approaches.
