Laser-wakefield accelerators (LWFAs) were proposed more than three decades ago, and while they promise to deliver compact, high energy particle accelerators, they will also provide the scientific community with novel light sources. In a LWFA, where an intense laser pulse focused onto a plasma forms an electromagnetic wave in its wake, electrons can be trapped and are now routinely accelerated to GeV energies. From terahertz radiation to gamma-rays, this article reviews light sources from relativistic electrons produced by LWFAs, and discusses their potential applications. Betatron motion, Compton scattering and undulators respectively produce x-rays or gamma-rays by oscillating relativistic electrons in the wakefield behind the laser pulse, a counter-propagating laser field, or a magnetic undulator. Other LWFA-based light sources include bremsstrahlung and terahertz radiation. We first evaluate the performance of each of these light sources, and compare them with more conventional approaches, including radio frequency accelerators or other laser-driven sources. We have then identified applications, which we discuss in details, in a broad range of fields: medical and biological applications, military, defense and industrial applications, and condensed matter and high energy density science.