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Quantum computers must achieve large-scale, fault-tolerant operation to deliver on their promise of transformational processing power [1-4]. This will require thousands or millions of high-fidelity quantum gates and similar numbers of qubits [5]. Demonstrations using neutral-atom qubits trapped and manipulated by lasers have shown that this modality can provide high two-qubit gate (CZ) fidelities and scalable operation [6-10]. However, the gates in these demonstrations are driven by lasers that do not resolve individual qubits, with universal computation enabled by physical mid-circuit shuttling of the qubits. This relatively slow operation will greatly extend runtimes for useful, large-scale computation. Here we demonstrate a universal neutral-atom quantum computer with gate rates limited by optical switching times, rather than shuttling, by individually addressing tightly focused laser beams at an array of single atoms. We achieve CZ fidelity of 99.35(4)% and local single qubit RZ gate fidelity of 99.902(8)%. Moreover, we demonstrate non-destructive readout of alkali-atom qubits with sub-percent loss, which boosts operational speed. This technique also enables us to measure 99.73(3)% CZ fidelity with atom-loss events excluded, which is a record among long lived neutral-atom qubits and highlights the path to higher fidelity and error correction. Our results represent a critical step towards large-scale, fault-tolerant neutral-atom quantum computers that can execute computations on practical timescales.