Materials with negative Poisson’s ratio (NPR), also known as auxetic materials, display exotic properties such as expansion (contraction) in all directions under uni-axial tension (compression). For their unique properties, these materials find a broad range of applications in robotic, structural, aerospace, and biomedical engineering. Understanding the finite strain deformation of and extracting or avoiding instabilities in auxetic materials under mechanical loads are essential concepts for their practical applications. This study has been divided into two parts. First, as a proof of concept, we study the wrinkling behavior of functionally-graded auxetic membranes under different loading conditions: (i) Wrinkling of thin and soft auxetic membranes under in-plane tensile loads; (ii) Wrinkling and limit-point instabilities during inflation of circular, cylindrical and rectangular auxetic membranes under uniform pressure loading. We show that spatial inhomogeneities of the Young modulus and of the Poisson ratio can be suitably tailored to produce non-trivial wrinkling patterns, with wrinkled regions that can appear, broaden, merge and eventually disappear again, as the magnitude of the applied traction is increased monotonically (see Figure 1a). To model wrinkling in a functionally-graded membrane, we employ tension field theory and implement it in MATHEMATICA and COMSOL. In the second part, we resort to experiments on 3D printed Thermoplastic Polyurethane (TPU) samples supported by numerical simulations in ABAQUS, to study the large deformation behaviour of functionally-graded thin rectangular auxetic materials (see Figure 1b). Experimental data from uniaxial and biaxial tension tests along with the Digital Image Correlation technique are used to measure the local strains and anisotropy in the samples. We use variations in local strains to model an effective Poisson function and anisotropic behaviour and come up with an accurate strain energy density function that predicts the finite strain deformation and post-buckling behaviour of TPU auxetic materials in ABAQUS.