Abstract: A rotary machine, usable as an engine or as a pump, has a toroidal cylinder (25) which is swept by continuously rotating disc-like piston (34). The cylinder has a double disc valve arrangement (42, 44) at one point, so that it can be divided into a compression side and an expansion side. However the piston has to pass the position of the valve during each revolution, and the valve opens briefly to allow this to happen. The compressed gas is forced into a storage chamber (62) outside the cylinder, and a valve (64) which is operated independently of the double disc valve arrangement (42, 44) is used to determine how much gas is admitted to the combustion chamber (25c), and when it is admitted.