By employing the Stroh formalism, a general solution satisfying the basic laws of two-dimensional linear anisotropic elasticity has been written in a complex variable formulation. To study the stress singularity, suitable stress functions have been assumed in the exponential form. The singular order near the anisotropic elastic composite wedge apex can then be found by satisfying the boundary conditions. Since there are many material constants and boundary conditions involved, the characteristic equation for the singular order usually becomes cumbersome or leaves in the form of a system of simultaneous algebraic equations. It is therefore difficult to get any important parameters to study the failure initiation of the composite wedges. Through a careful mathematical manipulation, a key matrix N^ that contains the information of material properties and wedge geometries has been found to be a dominant matrix for the determination of the singular order. A closed-form solution for the order of stress singularity is thus written in a simple form. Special cases such as the wedge corners, cracks, interfacial joints or cracks, a crack terminating at the interface, etc. can all be studied in a unified manner.