The frontal collisions of a laser beam with relativistic electrons result in Compton-backscattered photons. The energy of these photons is dependent on the laser and electron energy in the range from kilo-electron-volts to tens of mega-electron-volts. In a sufficiently narrow backscattering angle the photons are nearly monochromatic. Over the past 30 years there have been several attempts to produce photon beams by laser backscattering from relativistic electrons stored in magnetic ring structures. One aim is to produce photons in the high mega-electron-volt energy range with fluxes useful for nuclear physics research; another is to produce photons in the high kilo-electron-volt energy range, which would be useful for medical applications, such as coronary angiography or treatment of tumour. Our present interest is to investigate the possibility of using 34 keV to 10 MeV photon beams for applications in stereotactic functional radiosurgery. We foresee the possibility of neurosurgical operations through the intact skull with precise and effective destruction of deeply lying millimetre-sized targets with minimal effects on intervening structures, high reproducibility and good prediction of the results. Our paper presents: a Monte Carlo study of radiosurgery based on cross firing with 34 keV to 100 MeV photon beams and 200 and 580 MeV proton beams, a theoretical description of the kinematics of Compton backscattering and estimates of the backscattered photon flux from several combinations of laser cavities at Nd:YAG (1.17 eV) and CO2 (0.117 eV) laser energies and electron storage rings energies in the range 0.1-1.3 GeV. As examples, existing magnetic structures, such as the DA phi NE Accumulator in the lower energy range and the Trieste Synchrotron Light Source ELETTRA in the higher energy range have been utilized in the calculations. The Monte Carlo study has shown that radiosurgery with photon beams of energies in mega-electron-volt energy range enables precise destruction of deeply lying millimetre-sized targets with minimal effects on intervening structures. Its precision is comparable to that of radiosurgery with 200-580 MeV proton beam, but our hope is that radiosurgery with lower energy photon beams could be more precise and less expensive. An average dose of 200 Gy can be delivered to a target of diameter 2 mm at the centre of an 18 cm diameter phantom in 1 h using photon beams of fluences 7.3 x 10(10), 1.8 x 10(10), 6.5 x 10(8), 2.2 x 10(8), 8.6 x 10(7) and 7.8 x 10(6) photons per second at 34 keV, 100 keV, 1 MeV, 3 MeV, 10 MeV and 100 MeV per cross section of beam of 2 mm diameter, respectively. 34-100 keV photon beams were studied in the hope of finding a strong enhancement of their efficiency if a stable high-Z element were to be introduced into the target's DNA. It is shown that, with a low-energy ring running at about 0.4 GeV and a Nd:YAG laser, it would be possible to obtain the required 3 MeV photon beam flux to deliver the average dose within 1 h, assuming an average distance between the source and the target of about 5 m. With a similar machine used at about 1.3 GeV and a CO2 laser, a 3 MeV photon beam is obtained and the exposure time can be reduced to less than 1 min, assuming a roughly 10 m distance between source and target (here a beam angle of 0.1 mrad only had to be considered due to the larger angular energy and yield spread). With a lower electron energy of 138 MeV and a CO2 laser, a 34 keV photon beam can be produced. More than 45 h would be needed to deliver the same dose. We hope that this time could be shortened considerably if stable iodine were introduced into the target with the help of a DNA-seeking molecular carrier. In this case the geometrical precision would be further improved.