  We review theories of dark matter (DM) beyond the collisionless paradigm,
known as self-interacting dark matter (SIDM), and their observable implications
for astrophysical structure in the Universe. Self-interactions are motivated,
in part, due to the potential to explain long-standing (and more recent) small
scale structure observations that are in tension with collisionless cold DM
(CDM) predictions. Simple particle physics models for SIDM can provide a
universal explanation for these observations across a wide range of mass scales
spanning dwarf galaxies, low and high surface brightness spiral galaxies, and
clusters of galaxies. At the same time, SIDM leaves intact the success of
$\Lambda$CDM cosmology on large scales. This report covers the following
topics: (1) small scale structure issues, including the core-cusp problem, the
diversity problem for rotation curves, the missing satellites problem, and the
too-big-to-fail problem, as well as recent progress in hydrodynamical
simulations of galaxy formation; (2) N-body simulations for SIDM, including
implications for density profiles, halo shapes, substructure, and the interplay
between baryons and self-interactions; (3) semi-analytic Jeans-based methods
that provide a complementary approach for connecting particle models with
observations; (4) merging systems, such as cluster mergers (e.g., the Bullet
Cluster) and minor infalls, along with recent simulation results for mergers;
(5) particle physics models, including light mediator models and composite DM
models; and (6) complementary probes for SIDM, including indirect and direct
detection experiments, particle collider searches, and cosmological
observations. We provide a summary and critical look for all current
constraints on DM self-interactions and an outline for future directions.
