  This article gives an introduction to the realization of effective quantum
magnetism with ultracold molecules in an optical lattice, reviews experimental
and theoretical progress, and highlights future opportunities opened up by
ongoing experiments. Ultracold molecules offer capabilities that are otherwise
difficult or impossible to achieve in other effective spin systems, such as
long-ranged spin-spin interactions with controllable degrees of spatial and
spin anisotropy and favorable energy scales. Realizing quantum magnetism with
ultracold molecules provides access to rich many-body behaviors, including many
exotic phases of matter and interesting excitations and dynamics.
Far-from-equilibrium dynamics plays a key role in our exposition, just as it
did in recent ultracold molecule experiments realizing effective quantum
magnetism. In particular, we show that dynamical probes allow the observation
of correlated many-body spin physics, even in polar molecule gases that are not
quantum degenerate. After describing how quantum magnetism arises in ultracold
molecules and discussing recent observations of quantum magnetism with polar
molecules, we survey prospects for the future, ranging from immediate goals to
long-term visions.
