| Abstract :We introduce the Schroedinger Maps which can be thought as free solutions of the geometric Schroedinger equation. More exactly, while the classical Schroedinger equation is written for functions taking values in $\mathhb{C}$ (complex plane), the range of a Schroedinger Map is a manifold (with a special structure). We explain the importance of these Maps and what are the fundamental aspects one would like to understand about them. Then we focus on the particular case when the target manifold is $\mathbb{S}^2$ (the two dimensional sphere) and review the most recent results along with our contribution to the field. |