Solitons are relevant structures in nonlinear physics that appear in several contexts such as nonlinear optics and Bose-Einstein condensation. They are localized waves which travel without changing shape, even after eventual collisions between them. Specifically, dark solitons represent traveling local minima of the density (in contrast to local maxima for the bright ones).

In the talk, I will discuss some analytical results on the existence and properties of dark soliton solutions to the Nonlinear Schrödinger Equation, which is the most usual model for describing Bose-Einstein condensation and light propagation in optical fibers. I will focus on the more recent advances concerning the nonlocal version of the NSE, which is intended to capture nonlocal physical interactions. I will finish by presenting some open problems.