We propose and analyse metasurface-bound invisible (non-scattering) partially open cavities where the inside field distribution can be engineered. It is demonstrated both theoretically and experimentally that the cavities exhibit unidirectional invisibility at the operating frequency with enhanced or suppressed field at different positions inside the cavity volume. Several examples of applications of the designed cavities are proposed and analyzed, in particular, cloaking sensors and obstacles, enhancement of emission, and "invisible waveguides". The non-scattering mode excited in the proposed cavity is driven by the incident wave and resembles an ideal bound state in the continuum of electromagnetic frequency spectrum. In contrast to known bound states in the continuum, the mode can stay localized in the cavity infinitely long, provided that the incident wave illuminates the cavity.