We review noncommutative Poisson structures on affine and projective spaces over $\mathbb{C}$. 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 is a joint work with Vladimir Sokolov (Landau Institute for theoretical physics, Moscow, Russia).