As conventional electronics approaches its ultimate limits, novel concepts of fast quantum control have been sought after. Lightwave electronics – the foundation of attosecond science – has opened a new arena by utilizing the oscillating carrier wave of intense light pulses to control electrons faster than a cycle of light. We employ atomically strong terahertz electromagnetic pulses to accelerate electrons through the entire Brillouin zone of solids, drive quasiparticle collisions, and generate high-harmonic radiation as well as high-order sidebands. The unique band structures of topological insulators allow for all-ballistic and quasi-relativistic acceleration of Dirac quasiparticles over distances as large as 0.5 μm. In monolayers of transition metal dichalcogenides, we switch the electrons’ valley pseudospin, opening the door to subcycle valleytronics. Finally, we show that lightwave electronics can be combined with ultimate atomic spatial resolution in state-selective ultrafast scanning tunneling microscopy.