Recent advancements in head-mounted wearable technology are revolutionizing the field of biopotential measurement, but the integration of these technologies into practical, user-friendly devices remains challenging due to issues with design intrusiveness, comfort, and data privacy. To address these challenges, this paper presents GAPses, a novel smart glasses platform designed for unobtrusive, comfortable, and secure acquisition and processing of electroencephalography (EEG) and electrooculography (EOG) signals. We introduce a direct electrode-electronics interface with custom dry soft electrodes to enhance comfort for long wear. An integrated parallel ultra-low-power RISC-V processor (GAP9, Greenwaves Technologies) processes data at the edge, thereby eliminating the need for continuous data streaming through a wireless link, enhancing privacy, and increasing system reliability in adverse channel conditions. We demonstrate the broad applicability of the designed prototype through validation in a number of EEG-based interaction tasks, including alpha waves, steady-state visual evoked potential analysis, and motor movement classification. Furthermore, we demonstrate an EEG-based biometric subject recognition task, where we reach a sensitivity and specificity of 98.87% and 99.86% respectively, with only 8 EEG channels and an energy consumption per inference on the edge as low as 121 $μJ$. Moreover, in an EOG-based eye movement classification task, we reach an accuracy of 96.68% on 11 classes, resulting in an information transfer rate of 94.78 bit/min, which can be further increased to 161.43 bit/min by reducing the accuracy to 81.43%. The deployed implementation has an energy consumption of 40 $μJ$ per inference and a total system power of only 12.4 mW, of which only 1.61% is used for classification, allowing for continuous operation of more than 22 h with a small 75 mAh battery.