The physics of nonlinear Thomson scattering (NLTS) of intense laser light with electrons is discussed. In the weakly relativistic scattering regime, the polarization ellipticity of NLTS light is experimentally found to depend on both the polarization and intensity of the incident light. In this same regime, NLTS is theoretically found to be uniquely capable of measuring attosecond duration relativistic electron bunches. In the highly relativistic intensity regime (1021W/cm2), NLTS from a laser-wakefield-driven relativistic electron beam produces extremely high-order harmonics (>500) and energetic x-rays. Several experimental demonstrations of radiological applications of NLTS x-ray light are also presented.