This review discusses the interaction between single photons and single quantum dots in photonic nanostructures, focusing on quantum optics with excitons in quantum dots. The ability to engineer light-matter interaction strength in integrated photonic nanostructures enables fundamental quantum-electrodynamics experiments, such as spontaneous-emission control, modified Lamb shifts, and enhanced dipole-dipole interaction. Highly efficient single-photon sources and giant photon nonlinearities can be implemented, with applications in 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 centers, or atoms. The progress and future prospects of applications in quantum-information processing are considered. The review begins with an introduction to quantum electrodynamics (QED), which studies the interaction between light and matter at the most fundamental level. QED has proven to be remarkably precise and has led to significant experimental progress, including the demonstration of the Lamb shift. The development of tools for experimenting with single photons and single atoms started in the 1970s following the invention of the laser. The first experimental demonstration that an excited atom emits a single photon at a time was reported by Kimble et al. (1977), which marked the birth of experimental quantum optics. In parallel with the development of atomic QED, major research efforts have been focused on solid-state alternatives. Solid-state systems have the advantage that the elaborate experimental techniques needed for trapping and cooling single atoms are not required. Both the emitter and the optical environment can be engineered to enhance the photon-emitter coupling, leading to the development of solid-state QED. The first major breakthrough in solid-state QED was the discovery of photoluminescence from single self-assembled quantum dots embedded in GaAs. Since then, growth methods have developed tremendously, and quantum dots can be tailored to have excellent optical properties. The review discusses the structural and optical properties of quantum dots, including their growth and structural properties, excitons in quantum dots, the transition matrix element, multiexcitonic states, and optical properties of quantum dots. It also discusses the application of quantum dots in photonic nanostructures, including photonic crystals, photonic cavities, nanophotonic waveguides, and the role of fabrication imperfections. The review also discusses spontaneous emission of single photons from solid-state emitters in photonic nanostructures, resonance fluorescence from a quantum dot, quantum electrodynamics in nanophotonic waveguides, cavity quantum electrodynamics with single quantum dots, and photonic quantum-information processing. The review concludes with a discussion of the future prospects ofThis review discusses the interaction between single photons and single quantum dots in photonic nanostructures, focusing on quantum optics with excitons in quantum dots. The ability to engineer light-matter interaction strength in integrated photonic nanostructures enables fundamental quantum-electrodynamics experiments, such as spontaneous-emission control, modified Lamb shifts, and enhanced dipole-dipole interaction. Highly efficient single-photon sources and giant photon nonlinearities can be implemented, with applications in 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 centers, or atoms. The progress and future prospects of applications in quantum-information processing are considered. The review begins with an introduction to quantum electrodynamics (QED), which studies the interaction between light and matter at the most fundamental level. QED has proven to be remarkably precise and has led to significant experimental progress, including the demonstration of the Lamb shift. The development of tools for experimenting with single photons and single atoms started in the 1970s following the invention of the laser. The first experimental demonstration that an excited atom emits a single photon at a time was reported by Kimble et al. (1977), which marked the birth of experimental quantum optics. In parallel with the development of atomic QED, major research efforts have been focused on solid-state alternatives. Solid-state systems have the advantage that the elaborate experimental techniques needed for trapping and cooling single atoms are not required. Both the emitter and the optical environment can be engineered to enhance the photon-emitter coupling, leading to the development of solid-state QED. The first major breakthrough in solid-state QED was the discovery of photoluminescence from single self-assembled quantum dots embedded in GaAs. Since then, growth methods have developed tremendously, and quantum dots can be tailored to have excellent optical properties. The review discusses the structural and optical properties of quantum dots, including their growth and structural properties, excitons in quantum dots, the transition matrix element, multiexcitonic states, and optical properties of quantum dots. It also discusses the application of quantum dots in photonic nanostructures, including photonic crystals, photonic cavities, nanophotonic waveguides, and the role of fabrication imperfections. The review also discusses spontaneous emission of single photons from solid-state emitters in photonic nanostructures, resonance fluorescence from a quantum dot, quantum electrodynamics in nanophotonic waveguides, cavity quantum electrodynamics with single quantum dots, and photonic quantum-information processing. The review concludes with a discussion of the future prospects of