This paper presents structures with multiple equilibrium configurations arising from the combination of a state of prestress and unilateral contacts. A design problem is posed where preloaded elastic springs and unilateral constraints are embedded throughout a mechanism. The spring parameters are designed such that multiple target configurations are immobilized due to contact. In each of these configurations, the spring forces maintain compressive reaction forces, immobilizing the structure. Each immobilized configuration can rigidly resist perturbation forces up to some finite magnitude where contact is lost. Hence, this case of multiple configurations in equilibrium due to the combination of prestress and contact is referred to as multi-configuration rigidity. Two examples of structures exhibiting multi-configuration rigidity are presented. First, a four bar linkage with a single kinematic degree of freedom is used to introduce the concept. In the context of the linkage, multi-configuration rigidity is compared to multi-stability, exhibiting the key differences between the two concepts. Then, a 24-degree-of-freedom kirigami surface is presented that can morph between flat and spherical configurations, motivated by RF antenna applications. By embedding torsional springs and fold angle stops throughout the structure, flat and spherical configurations are made rigid. Actuation between the configurations can easily be achieved by snapping the structure between the rigid configurations.