  (Abridged) The growing role played by numerical N-body simulations in
cosmological studies as a fundamental connection between theoretical modeling
and direct observations has led to impressive advancements also in the
development and application of specific algorithms designed to probe a wide
range of Dark Energy scenarios. Over the last decade, a large number of
independent and complementary investigations have been carried out in the field
of Dark Energy N-body simulations, starting from the simplest case of
homogeneous Dark Energy models up to the recent development of highly
sophisticated iterative solvers for a variety of Modified Gravity theories. In
this Review - which is meant to be complementary to the general Review by
Kuhlen et al. published in this Volume - I will discuss the range of scenarios
for the cosmic acceleration that have been successfully investigated by means
of dedicated N-body simulations, and I will provide a broad summary of the main
results that have been obtained in this rather new research field. I will focus
the discussion on a few selected studies that have led to particularly
significant advancements in the field, and I will provide a comprehensive list
of references for a larger number of related works. Due to the vastness of the
topic, the discussion will not enter into the finest details of the different
implementations and will mainly focus on the outcomes of the various
simulations studies. Although quite recent, the field of Dark Energy
simulations has witnessed huge developments in the last few years, and
presently stands as a reliable approach to the investigation of the fundamental
nature of Dark Energy.
