  Recently, we developed a pair of meshless finite-volume Lagrangian methods
for hydrodynamics: the 'meshless finite mass' (MFM) and 'meshless finite
volume' (MFV) methods. These capture advantages of both smoothed-particle
hydrodynamics (SPH) and adaptive mesh-refinement (AMR) schemes. Here, we extend
these to include ideal magneto-hydrodynamics (MHD). The MHD equations are
second-order consistent and conservative. We augment these with a
divergence-cleaning scheme, which maintains div*B~0 to high accuracy. We
implement these in the code GIZMO, together with a state-of-the-art
implementation of SPH MHD. In every one of a large suite of test problems, the
new methods are competitive with moving-mesh and AMR schemes using constrained
transport (CT) to ensure div*B=0. They are able to correctly capture the growth
and structure of the magneto-rotational instability (MRI), MHD turbulence, and
the launching of magnetic jets, in some cases converging more rapidly than AMR
codes. Compared to SPH, the MFM/MFV methods exhibit proper convergence at fixed
neighbor number, sharper shock capturing, and dramatically reduced noise, div*B
errors, and diffusion. Still, 'modern' SPH is able to handle most of our tests,
at the cost of much larger kernels and 'by hand' adjustment of artificial
diffusion parameters. Compared to AMR, the new meshless methods exhibit
enhanced 'grid noise' but reduced advection errors and numerical diffusion,
velocity-independent errors, and superior angular momentum conservation and
coupling to N-body gravity solvers. As a result they converge more slowly on
some problems (involving smooth, slowly-moving flows) but more rapidly on
others (involving advection or rotation). In all cases, divergence-control
beyond the popular Powell 8-wave approach is necessary, or else all methods we
consider will systematically converge to unphysical solutions.
