The invention relates to a cylinder head (1) for an internal combustion engine having at least one fire-face-side first cooling chamber (5a) and one second cooling chamber (5b), which adjoins the first cooling chamber (5a) in the axial direction of the cylinder, the first and second cooling chambers (5a, 5b) being separated from each other by means of an intermediate plate (7), a central receptacle (4) being arranged for an injection nozzle or ignition device for each cylinder (Z), and the first and second cooling chambers (5a, 5b) being flow-connected to each other in the region of the central receptacle (4); and having at least two, preferably four gas exchange valve openings (2, 3) per cylinder (Z), the first cooling chamber (5a) having a radial cooling duct (11, 12, 13, 14) in the region of at least one valve bridge (20, 21, 22, 23) between two gas exchange valve openings (2, 3). In order to improve the flow activity in disadvantaged flow regions in the simplest possible manner, the radial cooling duct (11, 12, 13, 14) has at least one reduction in cross section (15, 16, 17, 18) in a region lying radially outside the valve bridge (20, 21, 22, 23), said region preferably being further away from the cylinder axis (18) than the center (2a, 3a) of at least one gas exchange valve opening (2, 3).