Magnons are the quanta of spin waves and transport angular momenta through magnetically ordered materials. They can be used to distribute and control on-chip GHz signals without charge flow, thereby avoiding Joule heating. Beyond multiplexed signal processing, filtering, and Boolean logic, they allow for hardware implementation of neural networks exploiting cascaded magnon scattering on the nanoscale. A game-changing boost is expected if nonvolatile magnon-signal storage and in-memory computation schemes become realistic. We outline recent progress in experimental research and micromagnetic modeling toward these goals before sketching remaining challenges.