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 discuss the Minkowski sum property of the generalized chain polytopes and explain the importance of this property for applications in the theory of Grassmann varieties. If time permits, we will also discuss the case of generalized marked chain polytopes. Partially based on a joint work with Wojciech Samotij.