Abstract: A pump for compression of air includes a first inner stationary cylinder (1) arranged a second displaceable cylinder (2). A piston (3) arranged for reciprocating motion together with the first cylinder (1) divides the interior space of the first cylinder (1) into a first and a second chamber (8,9). A third cylinder (15) surrounding the second cylinder (2) is formed with a fourth chamber (16) in communication, via a passage (17), with a third chamber (12) in the second cylinder (2) and, via an inlet valve (19), with the surrounding atmosphere. Displacement of the piston rod (4) in one direction forces the air in the third chamber (12) into the first and second chambers (8,9) with resulting pressure increase in said chambers (8,9). At the same time, air is sucked into the fourth chamber (16). Upon displacement of the piston rod (4) in the opposite direction, the air in the fourth chamber (16) is forced into the third chamber (12) and further into the first chamber (8).