We present a new type of wireless access infrastructure consisting of a fabric of dispersed electronic circuits and antennas that collectively function as a massive, distributed antenna array. We have chosen to name this new wireless infrastructure "RadioWeaves" and anticipate they can be integrated into indoor and outdoor walls, furniture, and other objects, rendering them a natural part of the environment. Technologically, RadioWeaves will deploy distributed arrays to create both favorable propagation and antenna array interaction. The technology leverages on the ideas of large-scale intelligent surfaces and cell-free wireless access. Offering close to the service connectivity and computing, new grades in energy efficiency, reliability, and low latency can be reached. The new concept moreover can be scaled up easily to offer a very high capacity in specific areas demanding so. In this paper we anticipate how two different demanding use cases can be served well by a dedicated RadioWeaves deployment: a crowd scenario and a highly reflective factory environment. A practical approach towards a RadioWeaves prototype, integrating dispersed electronics invisibly in a room environment, is introduced. We outline the many and diverse R&D challenges that need to be addressed to realize the great potential of the RadioWeaves technology.