Auxetic mechanical metamaterials are artificially architected materials that possess negative Poisson’s ratio, demonstrating transversal contracting deformation under external vertical compression loading. Their physical properties are mainly determined by spatial topological configurations. Traditionally, classical auxetic mechanical metamaterials exhibit relatively lower mechanical stiffness, compared to classic stretching dominated architectures. Nevertheless, in recent years, several novel auxetic mechanical metamaterials with high stiffness have been designed and proposed for energy absorption, load-bearing, and thermal-mechanical coupling applications. In this paper, mechanical design methods for designing auxetic structures with soft and stiff mechanical behavior are summarized and classified. For soft auxetic mechanical metamaterials, classic methods, such as using soft basic material, hierarchical design, tensile braided design, and curved ribs, are proposed. In comparison, for stiff auxetic mechanical metamaterials, design schemes, such as hard base material, hierarchical design, composite design, and adding additional load-bearing ribs, are proposed. Multi-functional applications of soft and stiff auxetic mechanical metamaterials are then reviewed. We hope this study could provide some guidelines for designing programmed auxetics with specified mechanical stiffness and deformation abilities according to demand.