This lecture covers the principles and applications of synchrotrons, x-ray free-electron lasers, and small-angle x-ray scattering. Topics include radius of gyration, Guinier and Porod regimes, polydispersity, and experimental details. The instructor discusses the calculation of radius of gyration for various objects, the Guinier regime for spheres, and the extended Guinier regime for rods. Additionally, Gaussian chains and Kratky plots are explained, along with the Porod regime and Porod's law. The lecture concludes with a discussion on polydispersity in scattering objects and experimental considerations such as evacuated flight paths.