A heat transfer structure and system including a matrix of tubes and spheres bonded together to provide a conduit for a first fluid such as water and a plurality of interconnected paths for a second fluid such as hot flue gas. The paths are made up of the spaces between the spheres such that the walls of the paths are portions of spherical surfaces. The total path length is made less than twenty times the average radius of curvature of the spherical surfaces and the spacing between adjacent tube elements is of the same order of magnitude as the average length of the paths. A water heater or boiler so constructed may be made to transfer substantially all the heat in a flue gas to water in an average path length of one inch or less.