  Recent experiments have demonstrated that light and matter can mix together
to an extreme degree, and previously uncharted regimes of light-matter
interactions are currently being explored in a variety of settings. The
so-called ultrastrong coupling (USC) regime is established when the
light-matter interaction energy is a comparable fraction of the bare
frequencies of the uncoupled systems. Furthermore, when the interaction
strengths become larger than the bare frequencies, the deep-strong coupling
(DSC) regime emerges. This article reviews advances in the field of the USC and
DSC regimes, in particular, for light modes confined in cavities interacting
with two-level systems. An overview is first provided on the theoretical
progress since the origins from the semiclassical Rabi model until recent
developments of the quantum Rabi model. Next, several key experimental results
from a variety of quantum platforms are described, including superconducting
circuits, semiconductor quantum wells, and other hybrid quantum systems.
Finally, anticipated applications are highlighted utilizing USC and DSC
regimes, including novel quantum optical phenomena, quantum simulation, and
quantum computation.
