(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.

63 pages, 8 figures. Invited Review for the special issue "The next decade in Dark Matter and Dark Energy" of the new Open Access journal "Physics of the Dark Universe". Comments are welcome