  Understanding physical properties of quantum emitters strongly interacting
with quantized electromagnetic modes is one of the primary goals in the
emergent field of waveguide quantum electrodynamics (QED). When the
light-matter coupling strength is comparable to or even exceeds energies of
elementary excitations, conventional approaches based on perturbative treatment
of light-matter interactions, two-level description of matter excitations, and
photon-number truncation are no longer sufficient. Here we study in and out of
equilibrium properties of waveguide QED in such nonperturbative regimes on the
basis of a comprehensive and rigorous theoretical approach using an asymptotic
decoupling unitary transformation. We uncover several surprising features
ranging from symmetry-protected many-body bound states in the continuum to
strong renormalization of the effective mass and potential; the latter may
explain recent experiments demonstrating cavity-induced changes in chemical
reactivity as well as enhancements of ferromagnetism or superconductivity. To
illustrate our general results with concrete examples, we use our formalism to
study a model of coupled cavity arrays, which is relevant to experiments in
superconducting qubits interacting with microwave resonators or atoms coupled
to photonic crystals. We examine the relation between our results and
delocalization-localization transition in the spin-boson model; notably, we
point out that a reentrant transition can occur in the regimes where the
coupling strength becomes the dominant energy scale. We also discuss
applications of our results to other problems in different fields, including
quantum optics, condensed matter physics, and quantum chemistry.
