Abstract: A rotary piston machine (64) of trochoidal construction with a piston (36) of epitrochoidal type with 1:1 generating circles and defined with an outer envelope and a follow-up mechanism (50) for the piston comprising a pair of relatively offset eccenters (52, 54) mounted for eccentric rotation about a drive shaft (14) with the piston, the eccenters (52, 54) being rotatably mounted in respective guide members (58, 60) which are constrained to reciprocate rectilinearly along angularly offset paths, the pair of eccenters (52, 54) being immediately adjacent the piston (36) and secured for rotation directly therewith. Where two or more pistons (36) are provided, each will have a respective follow-up mechanism (50). The piston (36) and follow-up mechanism (50) are mounted on an eccentric portion (30) of the drive shaft (14) which may be stepped with the eccenters (52, 54) being mounted about a reduced diameter portion (32). The eccenters (52, 54) may be separable from the piston (36).