A press fabric for use on the press section of a paper machine has a multilayer structure, and is woven according to techniques normally applied in the weaving of forming fabrics. The press fabric is woven from at least one system of warp yarns and two or more layers of weft yarns, with the yarn sizes and weave patterns chosen with a view toward providing the resulting press fabric with a fine paper-contacting surface, and ample void volume. One of the two or more layers of weft yarns includes, or is entirely composed of, multicomponent yarns having a plurality of load-bearing members. The multicomponent yarns may be multifilament or multistrand yarns composed of fine filaments, or may be polyurethane-coated monofilament yarns, and provide the press fabrics with a compressible and resilient structure. The multifilament or multistrand yarns may also be polyurethane-coated.