Abstract: Composite structures composed of inorganic membranes or polymer membranes supported on a multilayered woven wire mesh substrate are provided. Also provided are methods of making the composite structures and methods of using the composite structures as separation membranes. The mesh substrates are composed of a stack of two or more layers of woven wire mesh, wherein the different mesh layers in the stack have different mesh sizes. The multilayered mesh structure can support a defect-free, or substantially defect-free, membrane and has sufficient mechanical strength to allow the supported membranes to be used for chemical separations.