<p>The representation theory of tensor functions is essential to constitutive modeling of materials including both mechanical and physical behaviors. Generally, material symmetry is incorporated in the tensor functions through a structural or anisotropic tensor that characterizes the corresponding point group. The general mathematical framework was well-established in the 1990s. Nevertheless, the traditional theory suffers from a grand challenge that many point groups involve fourth or sixth order structural tensors that hinder its practical applications in engineering. Recently, researchers have reformulated the representation theory and opened up opportunities to model anisotropic materials using lower-order (i.e., 2nd- order and lower) structural tensors only, although the theory was not fully established. This work aims to fully establish the reformulated representation theory of tensor functions for all two-dimensional point groups. It was found that each point group needs a structural tensor set to characterize the symmetry. For each two-dimensional point group, the structural tensor set is proposed and the general tensor functions are derived. Only lower-order structural tensors are introduced so researchers can readily apply these tensor functions for their modeling applications. The theory presented here is useful for constitutive modeling of materials in general, especially for composites, nanomaterials, soft tissues, etc.</p>