Key properties of quantum materials stem from dynamic interaction chains that connect stable electronic quasiparticles through short-lived coherences, which are difficult to control at their natural time and length scales. Lightwave electronics sculpts the quantum flow of electrons and coherences faster than an oscillation cycle of light by using intense optical-carrier waves as fast biasing fields, which can access multi-electron interaction chains. In this Review, we summarize the key functionalities and the latest advances in lightwave electronics for both fundamental and technological explorations. For example, lightwave-driven ballistic electron transport through dynamically changing band structures has already led to the demonstration of phenomena such as high-harmonic emission and dynamic Bloch oscillations. Lightwave electronic control could also seamlessly convert quantum states between light and matter to create quantum chips that simultaneously exploit electronics for efficient interactions and optics for speed or long coherence lifetimes. Additionally, we present an outlook towards applications of lightwave electronics including quasiparticle colliders to explore quantum phenomena; all-optical band-structure reconstruction in ambient conditions; attoclocks to measure the interaction dynamics of diverse quantum phenomena; ultrafast electron videography to watch electronic reactions unfold; efficient light sources to create compact integration; and petahertz electronics to speed up traditional semiconductor electronics. Lightwave electronics could enable the control of interactions in quantum materials and provide access to the quantum phases and quantum information of condensed-matter systems. This Review discusses the fundamental concepts of lightwave electronics and outlines key advances and potential applications.