Abstract: Basic aero-thermodynamic equations of three-dimensional flow in turbomachine expressed with respect to non-orthogonal curvilinear coordinates and non-orthogonal velocity components were derived with tensor method in this paper. A number of special forms of these equations governing fluid flow on S1 and S2 relative stream surfaces are obtained corresponding to a number of special choice of the coordinates. Two groups of methods of solution suitable for inverse (design) and direct (analysis and off-design calculation) turbomachine problems were discussed. The first is based on the "mean streamline circumferential (or radial) extension" technique suggested by the present author. The second is essentially the matrix and iterative methods previously used for solving the stream-function equations which were expressed in terms of orthogonal coordinates. This new set of equations employing non-orthogonal curvilinear coordinates and non-orthogonal velocity components enables one to calculate accurately and conveniently the inverse and direct problems in turbomachines having any geometry, such as blades having curved leading and trailing edge contours situated in a curved passage.