Trapped atomic ions are standards for quantum information processing, serving as quantum memories, hosts of quantum gates in quantum computers and simulators, and nodes of quantum communication networks. Quantum bits based on trapped ions enjoy a rare combination of attributes: They have exquisite coherence properties, they can be prepared and measured with nearly 100% efficiency, and they are readily entangled with each other through the Coulomb interaction or remote photonic interconnects. The outstanding challenge is the scaling of trapped ions to hundreds or thousands of qubits and beyond, at which scale quantum processors can outperform their classical counterparts in certain applications. We review the latest progress and prospects in that effort, with the promise of advanced architectures and new technologies, such as microfabricated ion traps and integrated photonics.