In a preferred embodiment, the invention is a non-deflatable tire assembly having a hollow toroid-shaped flanged insert, a flexible body molded around the insert to form a tire, and at least one rim onto which is mounted the tire. The flange of the insert extends toward the central axis of the insert, and is used to attach the tire to the rim(s) via bolts through matching holes within the flange and rim(s). The flexible body may be made of a foam polyurethane matrix formed of a combination of polyol and polyisocyanate, while the flanged insert may be made of a high density polyethylene. A tough, flexible, and abrasive-resistant layer of polyurethane elastomer is preferably formed around the flexible body to define sidewalls and treads of the tire. To aid in the construction of the assembly and/or to better transmit torque and braking loads from an axle (to which the tire assembly is mounted) to the tire, mating protuberances and holes may be formed in the flange and the rim(s), the rims may be mounted to the tire with an interference fit, and/or the surface of the insert facing the flexible body molded around the insert may be ribbed. Optionally, and for light load conditions, the flange of the insert may extend to and be secured to the axle itself, omitting the need for rims in the tire assembly.