In this talk I will describe a number of recent results on the properties of elastic sheets and rings under compression, focusing on their wrinkling and buckling behavior. Wrinkling arises in substrate-supported systems owing to the competition between the forces exerted by the substrate and the elastic properties of the sheet or ring. This competition is
responsible for the presence of an intrinsic length scale in the system that is absent in pure buckling problems. I will show that the wrinkled states of a large class of substrate-supported sheets and rings under compression are described by an exactly solvable family of models. The resulting wrinkle profiles are shown to be related to the buckled states of the substrate-free sheet or ring and are therefore universal. Closed analytical expressions for the resulting universal shapes are provided, including the one-to-one relations between the pressure and tension at which these emerge. The analytical predictions agree with numerical continuation results to within numerical accuracy, for a large range of parameter values, up to the point of self-contact. Secondary bifurcations to mixed modes and to circumferentially localized fold states of a ring are identified and the resulting states, including chiral states, will also be described but these are no longer universal. Applications to the wrinkling of the inner lining of arteries undergoing systolic-diastolic pressure changes, and to a centrifugally driven Rayleigh-Taylor instability of a higher density fluid separated from a lower density fluid by an elastic interface will be emphasized.