  We extend recently-developed mesh-free Lagrangian methods for numerical
magnetohydrodynamics (MHD) to arbitrary anisotropic diffusion equations,
including: passive scalar diffusion, Spitzer-Braginskii conduction and
viscosity, cosmic ray diffusion/streaming, anisotropic radiation transport,
non-ideal MHD (Ohmic resistivity, ambipolar diffusion, the Hall effect), and
turbulent 'eddy diffusion.' We study these as implemented in the code GIZMO for
both new meshless finite-volume Godunov schemes (MFM/MFV). We show the MFM/MFV
methods are accurate and stable even with noisy fields and irregular particle
arrangements, and recover the correct behavior even in arbitrarily anisotropic
cases. They are competitive with state-of-the-art AMR/moving-mesh methods, and
can correctly treat anisotropic diffusion-driven instabilities (e.g. the MTI
and HBI, Hall MRI). We also develop a new scheme for stabilizing anisotropic
tensor-valued fluxes with high-order gradient estimators and non-linear flux
limiters, which is trivially generalized to AMR/moving-mesh codes. We also
present applications of some of these improvements for SPH, in the form of a
new integral-Godunov SPH formulation that adopts a moving-least squares
gradient estimator and introduces a flux-limited Riemann problem between
particles.
