The Stanley chain polytope of a finite poset P is defined by the set of inequalities labeled by the maximal chains in P. These polytopes have many nice properties and show up in various applications. We will
introduce a new class of polytopes, generalizing the case of chain polytopes.  The defining inequalities in our case are labeled by all subposets of P. We will describe combinatorial properties of the generalized chain polytopes and discuss applications in algebraic and geometric theory of Grassmann varieties. Partially based on a joint work with Wojciech Samotij.