  We discuss the properties of atom-photon bound states in waveguide QED
systems consisting of single or multiple atoms coupled strongly to a
finite-bandwidth photonic channel. Such bound states are formed by an atom and
a localized photonic excitation and represent the continuum analog of the
familiar dressed states in single-mode cavity QED. Here we present a detailed
analysis of the linear and nonlinear spectral features associated with single-
and multi-photon dressed states and show how the formation of bound states
affects the waveguide-mediated dipole-dipole interactions between separated
atoms. Our results provide a both qualitative and quantitative description of
the essential strong-coupling processes in waveguide QED systems, which are
currently being developed in the optical and the microwave regime.
