  This paper concerns the Cauchy problem of the two-dimensional (2D)
nonhomogeneous incompressible Magnetohydrodynamic (MHD) equations with vacuum
as far field density. We establish the global existence and uniqueness of
strong solutions to the 2D Cauchy problem on the whole space $\mathbb{R}^2$,
provided that the initial density and the initial magnetic decay not too slow
at infinity. In particular, the initial data can be arbitrarily large and the
initial density can contain vacuum states and even have compact support.
Furthermore, we also obtain the large time decay rates of the gradients of
velocity, magnetic and pressure.
