Contact discontinuities of the ideal compressible magnetohydrodynamics (MHD) are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic conservation laws. We prove the local existence and uniqueness of MHD contact discontinuities in both 2D and 3D in Sobolev spaces without any additional conditions, which in particular gives a complete answer to the two open questions raised by Morando, Trakhinin and Trebeschi, and there is no loss of derivatives in our well-posedness theory. The solution is constructed as the inviscid limit of solutions to suitably-chosen nonlinear approximate problems for the two-phase compressible viscous non-resistive MHD. This is a joint work with Professor Zhouping Xin (CUHK).