  The past two decades have seen a remarkable progress in the development of
synthetic colloidal agents which are capable of creating directed motion in an
unbiased environment at the microscale. These self-propelling particles are
often praised for their enormous potential to self-organize into dynamic
nonequilibrium structures such as living clusters, synchronized super-rotor
structures or self-propelling molecules featuring a complexity which is rarely
found outside of the living world. However, the precise mechanisms underlying
the formation and dynamics of many of these structures are still barely
understood, which is likely to hinge on the gaps in our understanding of how
active colloids interact. In particular, besides showing comparatively
short-ranged interactions which are well known from passive colloids (Van der
Waals, electrostatic etc.), active colloids show novel hydrodynamic
interactions as well as phoretic and substrate-mediated "osmotic"
cross-interactions which hinge on the action of the phoretic field gradients
which are induced by the colloids on other colloids in the system. The present
article discusses the complexity and the intriguing properties of these
interactions which in general are long-ranged, non-instantaneous, non-pairwise
and non-reciprocal and which may serve as key ingredients for the design of
future nonequilibrium colloidal materials. Besides providing a brief overview
on the state of the art of our understanding of these interactions a key aim of
this review is to emphasize open key questions and corresponding open
challenges.
