The generalized Landau-de Gennes theory for uniaxial and biaxial nematic liquid crystals is analyzed using tensor analysis. In this way all terms allowed by symmetry appear in the free-energy expression. This improves the quantitative shortcomings of the commonly used theory. It appears that uniaxial nematic liquid crystals are described by 12 independent generalized elastic terms and that biaxial nematic liquid crystals are described by 48 independent generalized elastic terms. For practical purposes, several approximations based on symmetry arguments are discussed. The theory is applied to describe the variation in order and orientation near a surface. The relevance of the theory to the debate on the surface elastic constant K 13 is pointed out.