  Photonic nanostructures provide means of tailoring the interaction between
light and matter and the past decade has witnessed a tremendous experimental
and theoretical progress in this subject. In particular, the combination with
semiconductor quantum dots has proven successful. This manuscript reviews
quantum optics with excitons in single quantum dots embedded in photonic
nanostructures. The ability to engineer the light-matter interaction strength
in integrated photonic nanostructures enables a range of fundamental
quantum-electrodynamics experiments on, e.g., spontaneous-emission control,
modified Lamb shifts, and enhanced dipole-dipole interaction. Furthermore,
highly efficient single-photon sources and giant photon nonlinearities may be
implemented with immediate applications for photonic quantum-information
processing. The review summarizes the general theoretical framework of photon
emission including the role of dephasing processes, and applies it to photonic
nanostructures of current interest, such as photonic-crystal cavities and
waveguides, dielectric nanowires, and plasmonic waveguides. The introduced
concepts are generally applicable in quantum nanophotonics and apply to a large
extent also to other quantum emitters, such as molecules, nitrogen vacancy
ceters, or atoms. Finally, the progress and future prospects of applications in
quantum-information processing are considered.
