An x-ray or gamma-ray source having: - an electron source configured in operation to generate a train of spaced electron bunches ; - an RF linear accelerator adapted to accelerate said electron bunches , which are directed along a path into a laser-electron beam interaction region; and - a laser system adapted to produce a laser pulse , the laser system is arranged so that the laser pulse is reflected into said path of the oncoming electron bunches to interact with the electron bunches, so that the laser-electron beam interaction region extends along said path, characterized in that : the transit time of each of the accelerated electron bunches through the laser-electron beam interaction region is both greater than the duration of the accelerated electron bunch and greater than the spacing between electron bunches; the laser system adapted to produce a laser pulse having a duration at least as long as a transit time of the laser pulse through the laser-electron beam interaction region, wherein the duration of the laser pulse is either: equal to the total length of the train of spaced electron bunches so that a single pass of the laser pulse through the laser-electron beam interaction region interacts with all of the accelerated electron bunches of the train; or some integer sub-multiple of the length of the train of the spaced electron bunches, and the laser system is arranged to recirculate the laser pulse through the laser-electron beam interaction region for a predetermined number of passes equal to an inverse of the sub-multiple; and thereby produce high flux beams of bright, tunable, polarized quasi-monoenergetic x-rays or gamma-rays via laser-Compton scattering. - an electron source configured in operation to generate a train of spaced electron bunches ; - an RF linear accelerator adapted to accelerate said electron bunches , which are directed along a path into a laser-electron beam interaction region; and - a laser system adapted to produce a laser pulse , the laser system is arranged so that the laser pulse is reflected into said path of the oncoming electron bunches to interact with the electron bunches, so that the laser-electron beam interaction region extends along said path, the transit time of each of the accelerated electron bunches through the laser-electron beam interaction region is both greater than the duration of the accelerated electron bunch and greater than the spacing between electron bunches; the laser system adapted to produce a laser pulse having a duration at least as long as a transit time of the laser pulse through the laser-electron beam interaction region, wherein the duration of the laser pulse is either: equal to the total length of the train of spaced electron bunches so that a single pass of the laser pulse through the laser-electron beam interaction region interacts with all of the accelerated electron bunches of the train; or some integer sub-multiple of the length of the train of the spaced electron bunches, and the laser system is arranged to recirculate the laser pulse through the laser-electron beam interaction region for a predetermined number of passes equal to an inverse of the sub-multiple; equal to the total length of the train of spaced electron bunches so that a single pass of the laser pulse through the laser-electron beam interaction region interacts with all of the accelerated electron bunches of the train; or some integer sub-multiple of the length of the train of the spaced electron bunches, and the laser system is arranged to recirculate the laser pulse through the laser-electron beam interaction region for a predetermined number of passes equal to an inverse of the sub-multiple; and thereby produce high flux beams of bright, tunable, polarized quasi-monoenergetic x-rays or gamma-rays via laser-Compton scattering. <EOS>