In a nuclear reactor having the fuel disposed within a large number of pressure tubes fed in parallel with liquid coolant which is partly vaporized while flowing through the pressure tubes, the invention provides a coolant-supply header assembly comprising parallel-extending sub-header pipes which at their ends are connected by end pipes and which have water outlets (for connection to the individual pressure tubes) distributed along their lengths, and header pipes each extending transversely across the sub-header pipes at spaced intervals therealong and having a water inlet and a plurality of branch pipes each of which connects to a respective one of the sub-header pipes.Preferably, a plurality of main coolant circuits are provided, each including such a header assembly, and the fuel in each pressure tube is cooled not only by the main coolant flow from one of the header assemblies but also by spray cooling utilizing coolant drawn from another of the coolant circuits, with each circuit maintaining a balance between coolant received as spray cooling in its own pressure tubes and coolant drawn off for spray cooling of other pressure tubes.