An array of trapped atomic ions that are laser cooled and isolated in an ultrahigh-vacuum environment presents one of the most advanced physical platforms for realizing a practical quantum computer. Small-scale ion quantum computers up to tens of ions have been built and achieved the highest fidelities on elementary quantum operations and overall quantum volume (QV). This article provides an overview of the elements of trapped-ion quantum computing (TIQC), current achievements in the field, and future perspectives.