Abstract - Understanding how mechanical properties of biological tissues emerge from cellular behavior is vital for understanding the mechanisms that guide embryonic development, cancer growth, and wound healing. Recently, a new type of rigidity transition was discovered in a family of vertex models for 2D and 3D tissues. Here I discuss these transitions and show that they are an instance of a much more general class of transitions in athermal, under-constrained systems. This kind of transition is also an important limiting case to understand the elastic properties of biopolymer networks like collagen. Under-constrained systems are generally floppy, however they can be rigidified by forcing them into a regime of geometric incompatibility. We show that these materials exhibit generic elastic behavior close to this transition, which is independent of the microscopic structure and the disorder in the system. Phrasing the condition of geometric incompatibility in terms of a minimal length function, we obtain analytic expressions for the elastic stresses and moduli. We numerically verify our findings by simulations of under-constrained spring networks as well as 2D and 3D vertex models for dense biological tissues. For instance, we analytically show that the ratio of the excess shear modulus to the shear stress is inversely proportional to the critical shear strain with a prefactor of three, which we expect to be a general hallmark of rigidity in under-constrained materials induced by geometric incompatibility. This could also be used in experiments to distinguish whether strain-stiffening as observed for instance in biopolymer networks arises from nonlinear characteristics of the microscopic material components or from effects of geometric incompatibility.