  In quantum mechanics, a fundamental law prevents quantum communications to
simultaneously achieve high rates and long distances. This limitation is well
known for point-to-point protocols, where two parties are directly connected by
a quantum channel, but not yet fully understood in protocols with quantum
repeaters. Here we solve this problem bounding the ultimate rates for
transmitting quantum information, entanglement and secret keys via quantum
repeaters. We derive single-letter upper bounds for the end-to-end capacities
achievable by the most general (adaptive) protocols of quantum and private
communication, from a single repeater chain to an arbitrarily-complex quantum
network, where systems may be routed through single or multiple paths. We
analytically establish these capacities under fundamental noise models,
including bosonic loss which is the most important for optical communications.
In this way, our results provide the ultimate benchmarks for testing the
optimal performance of repeater-assisted quantum communications.
