Abstract: A heat transfer system has a rotary machine 22 with compressor and expander regions and gas is first compressed in machine 22, passes through heat exchanger 23, is heated in combustor 24, then expands in machine 22, then passes through heat exchanger 23 to heat the gas, then passes through heat exchanger 107 to heat fluid in line 108. The machine 22 drives a heat pump 110 to heat fluid in line 111. Arrangements having two rotary machines are also described. A rotary machine has a rotor eccentrically mounted in a casing having axial end parts and a circumferential part and with vanes defining compartments with the casing and providing a compression region and an expansion region, valve means in the circumferential part adjacent the upstream edge of the outlet from one or both of the regions and responsive to pressure in the adjacent compartment to reduce or avoid excess pressure in the compression region or suction in the expansion region.