We show several results providing global fixed points for nilpotent groups of orientation-preserving $C^{1}$ diffeomorphisms of the plane ${\mathbb R}^{2}$. The main cases are namely groups of diffeomorphisms of the sphere such that $\infty$~is a global fixed point, groups of diffeomorphisms preserving a non-empty compact set and finally groups of diffeomorphisms preserving a probability measure.