The present invention provides monopolar acetabular liners (32, 54) that can enclose a femoral head (14) to form a hip replacement prosthesis (30, 50), wherein the liner (32, 54) has a rim (34) that creates an orifice, wherein the orifice has a diameter that is smaller than the diameter of the femoral head (14), and wherein the liner (32, 54) encompasses greater than 50 percent of the volume of the head (14), wherein a constraining ring (56) can be used in conjunction with the monopolar constrained acetabular liner (32, 54) to provide additional support for the femoral head (14) to stay with the acetabular liner (32, 54) and avoid dislocation. Assemblies and complete prosthetic replacements are also provided. Liners (32, 54) and constraining rings (56) having no, one or more recesses (36) also are provided.