In this paper, we introduce the concept of $b$-branchings in digraphs, which is a generalization of branchings serving as a counterpart of $b$-matchings. Here $b$ is a positive integer vector on the vertex set of a digraph, and a $b$-branching is defined as a common independent set of two matroids defined by $b$: an arc set is a $b$-branching if it has at most $b(v)$ arcs sharing the terminal vertex $v$, and it is an independent set of a certain sparsity matroid defined by $b$. We demonstrate that $b$-branchings yield an appropriate generalization of branchings by extending several classical results on branchings. We first present a multi-phase greedy algorithm for finding a maximum-weight $b$-branching. We then prove a packing theorem extending Edmonds' disjoint branchings theorem, and provide a strongly polynomial algorithm for finding optimal disjoint $b$-branchings. As a consequence of the packing theorem, we prove the integer decomposition property of the $b$-branching polytope. Finally, we deal with a further generalization in which a matroid constraint is imposed on the $b(v)$ arcs sharing the terminal vertex $v$.