The moduli spaces of polygons in $\mathbb{R}^3$, with prescribed side lengths, are naturally both symplectic and compact Kähler manifolds, and also fit in the framework of certain quiver representations. In this talk, we present some old and some new results on their topological classification, such as cohomological rigidity. Finally, we classify all the diffeomorphism types of polygon spaces which are Fano varieties and describe their toric geometry. This classification involves a very simple (and very rare) condition on the side lengths, and is joint work with Leonor Godinho (IST).