Precise and robust quantum control is the key to emerging quantum technologies. We uncover the essential correspondence between driven noisy quantum evolution and geometric curves described with the Frenet-Serret formulas. In terms of the geometric correspondence, we develop an explicit quantum robust control theory. The theory gives necessary and sufficient conditions of the robust control for given errors, and also offers quantitative measure of control robustness. Based on the analytic theory, we propose and demonstrate a practical framework to construct universal robust quantum gates for realistic qubits. We establish the analytic-numerical hybrid protocol to obtain arbitrary robust control pulses with simplest waveforms and arbitrary gate time. We obtain universal quantum gates with high fidelities above the fault tolerance threshold over a broad range of error strength. These results are tested numerically for realistic semiconductor spin qubits and superconducting transmon qubits, demonstrating the experimental feasibility of our theory.