We present measurements of x-ray emission from relativistic electrons passing through crystals and multilayer nanostructures mounted inside betatrons. Both spectra and yields have been measured. The measured spatial distributions and orientation dependencies are presented and are found to be in good agreement with theory. Betatrons developed over the past 30 years in Russia are compact and relatively inexpensive compared to LINACs and Storage Rings, and thus can be used in small laboratory settings. Various thin novel radiators mounted inside the betatron toroid can be used to generate (1) monochromatic tunable x-rays from crystalline and multilayer targets (2) tailored γ-ray emission spectra from single thin foils and (3) soft x-ray spectra from multiple thin foils (transition radiation). Although betatrons have relatively low current, the thin radiators permit the multipassing of the electron beam for increased efficiency.