The observed acceleration of the cosmic expansion represents one of
the most intriguing open problems of modern Physics. Although
observationally consistent with the effects of a cosmological
constant, its fundamental nature might reside in more complex physical
mechanisms like a slowly evolving light field or a modification of the
laws of gravity at large scales. From an observational point of view,
however, it is often extremely difficult to distinguish these
different possible Dark Energy models from the standard cosmological
constant if only background and linear observables are used. In this
context, the nonlinear regime of structure formation - directly
testable by means of large numerical simulations - offers the
possibility to break such degeneracy by highlighting possible
characteristic observational footprints of different competing Dark
Energy scenarios.

In this talk, I will provide a general overview of the field of Dark
Energy simulations, from the simplest case of homogeneous Quintessence
to the more complex case of inhomogeneous and interacting Dark Energy
and Modified Gravity simulations. I will then present some specific
examples of Dark Energy models that evade all background and linear
perturbations tests and that could be possibly identified only through
nonlinear observables. Finally, I will discuss how these different
numerical approaches will be refined and systematically employed in
the near future to drive observational efforts in the field of Dark
Energy investigations, with a particular focus on the planned
simulations program of the Euclid collaboration.