  Photon-photon scattering in vacuum is extremely weak. However, strong
effective interactions between single photons can be realized by employing
strong light-matter coupling. These interactions are a fundamental building
block for quantum optics, bringing many-body physics to the photonic world and
providing important resources for quantum photonic devices and for optical
metrology. In this Colloquium, we review the physics of strongly-interacting
photons in one-dimensional systems with no optical confinement along the
propagation direction. We focus on two recently-demonstrated experimental
realizations: superconducting qubits coupled to open transmission lines, and
interacting Rydberg atoms in a cold gas. Advancements in the theoretical
understanding of these systems are presented in complementary formalisms and
compared to experimental results. The experimental achievements are summarized
alongside a description of the quantum optical effects and quantum devices
emerging from them.
