  This thesis presents an introduction to the class of Richardson-Gaudin
integrable models, with special focus on the Bethe ansatz wave function, and
investigates ways of applying the properties of Richardson-Gaudin models both
in and out of integrability. A framework is outlined for the numerical and
theoretical treatment of these systems, exposing a duality allowing the Bethe
equations to be solved numerically. This is extended to the calculation of
inner products and correlation functions. Using this framework, the influence
of particle exchange on the Bethe ansatz is discussed, after which it is shown
how the Bethe ansatz is able to accurately model wave functions of
non-integrable models in two different settings. First, a variational approach
is outlined for stationary models where integrability-breaking perturbations
are explicitly introduced. Second, an alternative way of breaking integrability
is through the introduction of dynamics and periodic driving, where it is shown
how integrability can be used to model the resulting Floquet many-body
resonances. Throughout this work, it is shown how the clear-cut structure and
relatively large freedom in Richardson-Gaudin models makes them ideal for an
investigation of the general principles of integrability, as well as being a
perfect testing ground for the development of new quantum many-body techniques
beyond integrability.
