This paper presents a new asynchronous hybrid brain-computer interface (BCI) system that integrates a speller, a web browser, an e-mail client, and a file explorer using electroencephalographic (EEG) and electrooculography (EOG) signals. More specifically, an EOG-based button selection method, which requires the user to blink his/her eyes synchronously with the target button's flashes during button selection, is first presented. Next, we propose a mouse control method by combining EEG and EOG signals, in which the left-/right-hand motor imagery (MI)-related EEG is used to control the horizontal movement of the mouse and the blink-related EOG is used to control the vertical movement of the mouse and to select/reject a target. These two methods are further combined to develop the integrated hybrid BCI system. With the hybrid BCI, users can input text, access the internet, communicate with others via e-mail, and manage files in their computer using only EEG and EOG without any body movements. Ten healthy subjects participated in a comprehensive online experiment, and superior performance was achieved compared with our previously developed P300- and MI-based BCI and some other asynchronous BCIs, therefore demonstrating the system's effectiveness.