The high-pressure pump, for fuel for an engine, has a radial arrangement of pistons and cylinders. The pump has a fixed housing containing a rotating shaft which actuates the pistons. The outer ends of the cylinders are covered by a C-shaped cover plate (38) which screws into a female thread in a large-diameter opening in the cylinder block. The inlet valve (48) is fitted in the cylinder head and the low-pressure fuel flows through the space under the cover to the valve. The high-pressure outlet (52) is to the side of the cylinder, via a non-return valve (62).