【Abstract】Mu-bases, in another name of moving curves and moving surfaces, provide a bridge between the parametric forms and implicit forms of curves and surfaces. Mu-bases are new applications of the Syzygy module theory in geometric computing. THey ease many computational problems related to curves and surfaces by decompositing them to the intersection of moving lines or moving planes. We shall review the developments of mu-bases over the past decades, and show their power in sloving  problems of implicitization, parameterization, singularity computation, closed inversion formula, etc.