Brain–computer interfaces (BCIs) translate neural signals into control commands to restore or augment human capabilities, enabling robotic assistance for essential daily activities. Compared with intracortical BCIs, non-invasive BCIs require less medical and surgical intervention, but have yet to demonstrate reliable performance in complex everyday tasks with high success and low mental effort. We present Electroencephalography (EEG)-based Neural Signal Operated Intelligent Robots (NOIR-EEG), a general-purpose, intelligent, non-invasive BCI framework that allows users to command robots via EEG signals. NOIR-EEG combines advances in neural signal decoding with recent progress in AI and robotics, including large pre-trained models and intention learning. In tests, sixteen participants successfully completed fifteen challenging household tasks. Intention learning algorithms adapt to individual users and predict their goals, substantially reducing human effort.