Particle accelerators operate at pressures of about 10-7 Torr. Storage rings require 10-9 to 10-11 Torr. The maximum permissible pressure is determined by many perturbing interactions between beam particles and gas molecules. Proton storage rings have additional problems from interactions with layers of gas adsorbed on the chamber walls. Most accelerators and storage rings have a tubular structure with pumps connected in regular intervals. The pressure behaviour in such a system can be calculated from a differential equation, which can be extended to include beam-wall interactions, readsorption and distributed pumps. The systems are pumped by sputter-ion pumps, pre-evacuated by turbopumps and controlled by computers. Radioactivity poses special problems.