Appears in collection : Mathematics on the Crossroad of Centuries - A Conference in Honor of Maxim Kontsevich's 60th Birthday

We review noncommutative Poisson structures on affine and projective spaces over ${\mathbb C}$. This part of the talk is based on ideas of Maxim Kontsevich from his paper "Formal non-commutative symplectic geometry". We also construct a class of examples of noncommutative Poisson structures on ${\mathbb C} P^{n-1}$ for $n>2$. These noncommutative Poisson structures depend on a modular parameter $\tau\in{\mathbb C}$ and an additional descrete parameter $k\in{\mathbb Z}$, where $1 \leq k < n$ and $k,n$ are coprime. The abelianization of these Poisson structures can be lifted to the quadratic elliptic Poisson algebras $q_{n,k}(\tau)$. This talk is based on a joint paper with Vladimir Sokolov.