  Compressible vortex sheets are fundamental waves in entropy solutions to the
multidimensional hyperbolic systems of conservation laws. For the Euler
equations in 2-D gas dynamics, the classical linearized stability analysis on
compressible vortex sheets predicts stability when the Mach number $M>\sqrt{2}$
and instability when $M<\sqrt{2}$; and Artola-Majda's analysis reveals that the
nonlinear instability may occur if planar vortex sheets are perturbed by highly
oscillatory waves even when $M>\sqrt{2}$. For the Euler equations in 3-D, every
compressible vortex sheet is violently unstable and this violent instability is
the analogue of the Kelvin-Helmholtz instability for incompressible fluids. The
purpose of this paper is to understand whether compressible vortex sheets in
3-D, which are unstable in the regime of pure gas dynamics, become stable under
the magnetic effect in 3-D magnetohydrodynamics (MHD). One of the main features
is that the stability problem is equivalent to a free boundary problem whose
free boundary is a characteristic surface. Another feature is that the
linearized problem for current-vortex sheets in MHD does not meet the uniform
Kreiss-Lopatinskii condition. In this paper, we develop a nonlinear approach to
deal with these difficulties in 3-D MHD. We first carefully formulate the
linearized problem for the current-vortex sheets to show rigorously that the
magnetic effect makes the problem weakly stable and establish energy estimates,
especially high-order energy estimates, in terms of the nonhomogeneous terms
and variable coefficients without loss of the order. Then we exploit these
results to develop a suitable iteration scheme of Nash-Moser-H\"{o}rmander type
and establish its convergence, which leads to the existence and stability of
compressible current-vortex sheets, locally in time, in the 3-D MHD.
