AbstractIn this article, we introduce the b‐bibranching problem in digraphs, which is a common generalization of the bibranching and b‐branching problems. The bibranching problem, introduced by Schrijver, is a common generalization of the branching and bipartite edge cover problems. Previous results on bibranchings include polynomial algorithms, a linear programming formulation with total dual integrality, a packing theorem, and an M‐convex submodular flow formulation. The b‐branching problem, recently introduced by Kakimura, Kamiyama, and Takazawa, is a generalization of the branching problem admitting higher indegree, that is, each vertex v can have indegree at most b(v). For b‐branchings, a combinatorial algorithm, a linear programming formulation with total dual integrality, and a packing theorem for branchings are extended. A main contribution of this article is to extend those previous results on bibranchings and b‐branchings to b‐bibranchings. That is, we present a linear programming formulation with total dual integrality, a packing theorem, and an M‐convex submodular flow formulation for b‐bibranchings. In particular, the linear program and M‐convex submodular flow formulations, respectively, imply polynomial algorithms for finding a shortest b‐bibranching.