  We prove Taylor's conjecture which says that in 3D MHD, magnetic helicity is
conserved in the ideal limit in bounded, simply connected, perfectly conducting
domains. When the domain is multiply connected, magnetic helicity depends on
the vector potential of the magnetic field. In that setting we show that
magnetic helicity is conserved for a large and natural class of vector
potentials but not in general for all vector potentials. As an analogue of
Taylor's conjecture in 2D, we show that mean square magnetic potential is
conserved in the ideal limit, even in multiply connected domains.
