  The development of methods and algorithms to solve the $N$-body problem for
classical, collisionless, non-relativistic particles has made it possible to
follow the growth and evolution of cosmic dark matter structures over most of
the Universe's history. In the best studied case $-$ the cold dark matter or
CDM model $-$ the dark matter is assumed to consist of elementary particles
that had negligible thermal velocities at early times. Progress over the past
three decades has led to a nearly complete description of the assembly,
structure and spatial distribution of dark matter haloes, and their
substructure in this model, over almost the entire mass range of astronomical
objects. On scales of galaxies and above, predictions from this standard CDM
model have been shown to provide a remarkably good match to a wide variety of
astronomical data over a large range of epochs, from the temperature structure
of the cosmic background radiation to the large-scale distribution of galaxies.
The frontier in this field has shifted to the relatively unexplored subgalactic
scales, the domain of the central regions of massive haloes, and that of
low-mass haloes and subhaloes, where potentially fundamental questions remain.
Answering them may require: (i) the effect of known but uncertain baryonic
processes (involving gas and stars), and/or (ii) alternative models with new
dark matter physics. Here we present a review of the field, focusing on our
current understanding of dark matter structure from $N$-body simulations and on
the challenges ahead.
