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This paper proposes several axiomatic refined theories for the linear static analysis of beams made of materials whose properties are graded along one or two directions. Via a unified formulation, a generic N-order approximation is assumed for the displacement unknown variables over the beam cross-section. The governing differential equations and the boundary conditions are derived in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier type, closed form solution is adopted. Classical beam theories, such as Euler-Bernoulli's and Timoshenko's, are obtained as particular cases. Beams that undergo bending and torsional loadings are investigated. Several values of the span-to-height ratio are considered. Slender as well as deep beams are, therefore, investigated. Comparisons with elasticity solutions and three-dimensional finite element models are given. The numerical investigation shows that the proposed unified formulation yields the complete three-dimensional displacement and stress fields as long as the appropriate approximation order is considered. The accuracy of the solution depends upon the geometrical parameters of the beam and the loading conditions.