The pneumatic tire has proven to be the dominant design for rolling transport since shortly after its invention by John Boyd Dunlop in 1888. This paper identifies four critical characteristics that led to the success of the pneumatic tire: low energy loss on rough surfaces, low vertical stiffness, low contact pressure, and low mass. A non-pneumatic structure is proposed that exhibits these four critical characteristics while breaking some of the most restrictive design constraints imposed by pneumatic mechanics. This structure consists of a circular beam composed of two inextensible membranes separated by a relatively low modulus elastic material. The beam is connected to a hub by thin elastic spokes. This structure forming an integrated tire∕wheel is called a Tweel™.