  Quantum cascade lasers are unipolar semiconductor lasers covering a wide
range of the infrared and terahertz spectrum. Lasing action is achieved by
using optical intersubband transitions between quantized states in specifically
designed multiple-quantum-well heterostructures. A systematic improvement of
quantum cascade lasers with respect to operating temperature, efficiency and
spectral range requires detailed modeling of the underlying physical processes
in these structures. Moreover, the quantum cascade laser constitutes a
versatile model device for the development and improvement of simulation
techniques in nano- and optoelectronics. This review provides a comprehensive
survey and discussion of the modeling techniques used for the simulation of
quantum cascade lasers. The main focus is on the modeling of carrier transport
in the nanostructured gain medium, while the simulation of the optical cavity
is covered at a more basic level. Specifically, the transfer matrix and finite
difference methods for solving the one-dimensional Schr\"odinger equation and
Schr\"odinger-Poisson system are discussed, providing the quantized states in
the multiple-quantum-well active region. The modeling of the optical cavity is
covered with a focus on basic waveguide resonator structures. Furthermore,
various carrier transport simulation methods are discussed, ranging from basic
empirical approaches to advanced self-consistent techniques. The methods
include empirical rate equation and related Maxwell-Bloch equation approaches,
self-consistent rate equation and ensemble Monte Carlo methods, as well as
quantum transport approaches, in particular the density matrix and
non-equilibrium Green's function (NEGF) formalism. The derived scattering rates
and self-energies are generally valid for n-type devices based on
one-dimensional quantum confinement, such as quantum well structures.
