Pulses of x-rays 300 femtoseconds in duration at a wavelength of 0.4 angstroms (30,000 electron volts) have been generated by 90 degrees Thomson scattering between infrared tera-watt laser pulses and highly relativistic electrons from an accelerator, In the right-angle scattering geometry, the duration of the x-ray burst is determined by the transit time of the laser pulse across the similar to 90-micrometer waist of the focused electron beam, The x-rays are highly directed (similar to 0.6 degrees divergence) and can be tuned in energy. This source of femtosecond x-rays will make it possible to combine x-ray techniques with ultrafast time resolution to investigate structural dynamics in condensed matter.