Abstract: This paper presents some recent research work and advances on systematic methodology for elasticity, (i) A new orthogonality relationship is firstly proposed, and the related system of dual equations can be derived out directly without the concept of symplectic geometry; (ii) Based on the new orthogonality relationship, a direct solution method by eigenfunction expansion is established for 2-D elasticity, and thus, the explicit close solutions under diagonalizable boundary conditions can be obtained; (iii) The dual systematic methodology is generalized to multi-directions of coordinates, i.e., the dual differential equation and its solution system are established for multi-directions of coordinates; (iv) The bending theory of elastic plate domain is developed by the operator method for partial differential equations, and the solutions of this theory are decomposed into bending homogeneous solutions, particular solutions and attenuation solutions; (v) The dual systematic methodology is also generalized to the problems of thin and thick plates, related dual differential equations, orthogonality relationships and variational principles are successfully established.