  The advent of visible-infrared laser pulses carrying a substantial fraction
of their energy in a single field oscillation cycle has opened a new era in the
experimental investigation of ultrafast processes in semiconductors and
dielectrics (bulk as well as nanostructured), motivated by the quest for the
ultimate frontiers of electron-based signal metrology and processing. Exploring
ways to approach those frontiers requires insight into the physics underlying
the interaction of strong high-frequency (optical) fields with electrons moving
in periodic potentials. This Colloquium aims at providing this insight.
Introduction to the foundations of strong-field phenomena defines and compares
regimes of field--matter interaction in periodic systems, including (perfect)
crystals as well as optical and semiconductor superlattices, followed by a
review of recent experimental advances in the study of strong-field dynamics in
crystals and nanostructures. Avenues toward measuring and controlling
electronic processes up to petahertz frequencies are discussed.
