  The Bethe Ansatz is a method that is used in quantum integrable models in
order to solve them explicitly. This method is explained here in a general
framework, which applies to 1D quantum spin chains, 2D statistical lattice
models (vertex models) and relativistic field theories with 1 space dimension
and 1 time dimension. The connection with quantum groups is expounded. Several
applications are then presented. Finite size corrections are calculated via two
methods: The Non-Linear Integral Equations, which are applied to the study of
the states of the affine Toda model with imaginary coupling, and their
interpolation between the high energy (ultra-violet) and low energy (infra-red)
regions; and the Thermodynamic Bethe Ansatz Equations, along with the
associated Fusion Equations, which are used to determine the thermodynamic
properties of the generalized multi-channel Kondo model. The latter is then
studied in more detail, still using the Bethe Ansatz and quantum groups, so as
to characterize the spectrum of the low energy excitations.
