A basic paradigm underlying the Hookean mechanics of amorphous, isotropic solids is that small deformations are proportional to the magnitude of external forces. However, slender bodies may undergo large deformations even under minute forces, leading to nonlinear responses rooted in purely geometric effects. Here we identify stiffening and softening as two prominent motifs of such a material-independent, geometrically-nonlinear response of thin sheets, and we show how both effects emanate from nontrivial yet generic features of the stress and displacement fields. Our insights are borne out of studying the indentation of a polymer film on a liquid bath, using experiments, simulations, and theory. We find that stiffening is due to growing anisotropy of the stress field whereas softening is due to changes in shape. We discuss similarities with the mechanics of fiber networks, suggesting that our results are relevant to a wide range of two-and three-dimensional materials.A major challenge in the mechanics of materials and structures is bridging the gap between a system's local material response and its global stiffness. The connection from microscopic to macroscopic scales is often complicated by subtle geometric effects [1]. One conceptually simple example is given by an elastic rod, which may drastically change its shape in response to loading; such elastica problems captured the attention of Galileo, the Bernoullis, and Euler, and variations on it continue to fascinate and push our understanding of slender bodies today [2][3][4]. A twodimensional sheet may also carry tensile loads in one direction while buckling in a perpendicular direction, leading to large anisotropies that control the transmission of forces through the material [5,6]. Understanding such geometrically-nonlinear behaviors is important to a wide range of applications involving thin sheets, from stretchable electronics [7] to large-scale inflatable structures [8]. There is also a diverse array of structures that are essentially composed of many connected rods or sheets, from buckling-or origami-based mechanical metamaterials [9] to soft-robotic systems [10] to disordered networks of elastic fibers including biopolymer gels and the cytoskeleton [11,12]. While much work has been carried out to understand specific systems, much less is known about generic mechanisms underlying their global mechanical response.Here we identify two general types of geometrically-nonlinear behaviors by focusing on the response of a floating thin film to indentation [ Fig. 1a,b], using experiments and simulations spanning four decades in indentation depth and theory that addresses small and large deflections. In the first behavior, the onset of anisotropy in the stress field leads to more effective transmission of forces away from a point of loading, causing the observed force to stiffen (i.e., F/δ increases where F is the force due to the imposed displacement δ) [13]. In the second behavior, the geometry of the deformations causes the transmitted force to saturate at lar...