  The recent progress in the analytical solution of models invented to describe
theoretically the interaction of matter with light on an atomic scale is
reviewed. The methods employ the classical theory of linear differential
equations in the complex domain (Fuchsian equations). The linking concept is
provided by the Bargmann Hilbert space of analytic functions, which is
isomorphic to $L^2(\mathbb{R})$, the standard Hilbert space for a single
continuous degree of freedom in quantum mechanics. I give the solution of the
quantum Rabi model in some detail and sketch the solution of its
generalization, the asymmetric Dicke model. Characteristic properties of the
respective spectra are derived directly from the singularity structure of the
corresponding system of differential equations.
