Reconfigurable Intelligent Surfaces (RISs) are programmable metasurfaces that can adaptively steer received electromagnetic energy in desired directions by employing controllable phase shifting cells. Among other uses, an RIS can modify the propagation environment in order to provide wireless access to user locations that are not otherwise reachable by a base station. Alternatively, an RIS can steer the waves away from particular locations in space, to eliminate interference and allow for co-existence of the wireless network with other types of fixed wireless services (e.g., radars, unlicensed radio bands, etc.). The novel approach in this work is a wave-controlled architecture that properly accounts for the maximum possible change in the local reflection phase that can be achieved by adjacent RIS elements. It obviates the need for dense wiring and signal paths that would be required for individual control of every RIS element, and thus offers a substantial reduction in the required hardware. We specify this wave-controlled RIS architecture in detail and discuss signal processing and machine learning methods that exploit it in both point-to-point and multicell MIMO systems. Such implementations can lead to a dramatic improvement in next-generation wireless, radar, and navigation systems where RIS finds wide applications. They have the potential to improve the efficiency of spectrum utilization and coexistence by orders of magnitude.