The standard coordinate systems (i.e. cartesian, cylindrical-polar, spherical-polar or combinations theoreof) which are usually employed in the numerical solution of the transport differential equations, are suitable for simple geometries only. In the present study the momentum, mass and energy conservation equations are expressed in terms of general curvilinear-orthogonal coordinates, which enable the irregularly shaped solution domains, encountered in practice, to be completely mapped i.e. all boundaries to coincide with, coordinate surfaces. A mixed coordinate system is also introduced, which, is curvilinear-orthogonal in, the two of the three directions and rectilinear in the third direction. The latter system gives simpler equations and is suitable for straight flow passages of arbitrary cross sectional shape. The momentum, mass and energy conservation, differential, equations are transformed to finite-difference ones by integration over six-sided control volumes formed by coordinate surfaces and are then solved by an iterative procedure. The method is tested successfully in various flow and heat transfer cases.