This work presents an advanced study of the design of auxetic piezoelectric structures using topology and shape optimization. Auxetic piezoelectric structures exhibit high hydrostatic coupling coefficients, which makes them unique for a number of engineering applications, including actuation and sensing [1]. The performance characteristics of such materials are quite often determined using experimental studies [2]. Thus, we  explore the design possibilities of such metamaterials by employing structural optimization methods. Initial works by [3] and [4] have demonstrated the applicability of the topology optimization method for maximization of, e.g., hydrostatic piezoelectric coupling coefficients. In this work, we extend the study towards consideration of various stiffness behaviors in the auxetic structures, by separately controlling the effective elasticity constants of the final design. We demonstrate that the selection of material constraints greatly influences the final layout of the auxetic structure. Additionally, a curvature constraint can be imposed by employing node-based shape optimization to fine tune the design [5]. We show that both topology and shape optimization can be utilized sequentially for the design of metamaterials with specific geometrical and stiffness requirements. [1] Iyer, S., Alkhader, M., \& Venkatesh, T. A. (2015). Electromechanical behavior of auxetic piezoelectric cellular solids. Scripta Materialia, 99, 65-68. [2] Fey, T., Eichhorn, F., Han, G., Ebert, K., Wegener, M., Roosen, A., ... \& Greil, P. (2015). Mechanical and electrical strain response of a piezoelectric auxetic PZT lattice structure. Smart Materials and Structures, 25(1), 015017. [3] Sigmund, O., Torquato, S., \& Aksay, I. A. (1998). On the design of 1–3 piezocomposites using topology optimization. Journal of materials research, 13(4), 1038-1048. [4] Silva, E. N., Fonseca, J. O., \& Kikuchi, N. (1997). Optimal design of piezoelectric microstructures. Computational mechanics, 19(5), 397-410. [5] Stankiewicz, G., Dev, C., \& Steinmann, P. (2022). Geometrically nonlinear design of compliant mechanisms: Topology and shape optimization with stress and curvature constraints. Computer Methods in Applied Mechanics and Engineering, 397, 115161.