<p>This paper presents a contact boundary-based finite element method (BFEM) for solving the indentation behavior of multilayered and functionally graded anisotropic elastic plates subjected to rigid punches. The contact-BFEM divides the multilayered or functionally graded plates into different sublayers. Each sublayer is discretized following the standard rule of the boundary element method and assembled by the rule of the finite element. Thus, BFEM requires discretizing only the plate boundaries, making it well-suited for contact problems. By treating the rigid body motion of the punch as an additional variable, no mesh is required for the punch, which improves modeling and computational efficiency. The method is applicable to frictional contact and punches with arbitrary profiles. To illustrate the correctness and versatility of the method, numerical examples are presented. The results obtained by the contact-BFEM are compared with established boundary element solutions for benchmark problems such as half-plane and bi-material cases. For functionally graded materials, convergence studies are performed. Parametric studies are also conducted to study the influence of material gradation, anisotropy, surface friction, and punch profile on the contact responses.</p>