The quantum Rabi model, which describes the simplest interaction between light and matter, is a fundamental model with widespread applications in quantum physics. These include the interaction between light and trapped ions or quantum dots, and between microwaves and superconducting qubits.

The Rabi model is also applicable to both cavity and circuit QED.

Despite its simplicity, the eigenspectrum of the fully quantized version of the Rabi model has only recently been obtained analytically.

Moreover, the quantum Rabi model has been claimed to be integrable based on a criterion of quantum integrability without presupposing the existence of a set of commuting operators. In this talk I will discuss the solution of the Rabi model and the prospects for Yang-Baxter integrability. The more general Dicke model will also be discussed.