  We initiate the study of a class of polytopes, which we coin polypositroids,
defined to be those polytopes that are simultaneously generalized permutohedra
(or polymatroids) and alcoved polytopes. Whereas positroids are the matroids
arising from the totally nonnegative Grassmannian, polypositroids are
"positive" polymatroids. We parametrize polypositroids using Coxeter necklaces
and balanced graphs, and describe the cone of polypositroids by extremal rays
and facet inequalities. We introduce a notion of (W,c)-polypositroid for a
finite Weyl group W and a choice of Coxeter element c. We connect the theory of
(W,c)-polypositroids to cluster algebras of finite type and to generalized
associahedra. We discuss membranes, which are certain triangulated
2-dimensional surfaces inside polypositroids. Membranes extend the notion of
plabic graphs from positroids to polypositroids.
