Perhaps the most important aspect of contemporary condensed matter physics involves understanding strong Coulomb interactions between the large number of electrons in a solid. Electronic correlations lead to the emergence of new system properties, such as metal-insulator transitions, superconductivity, magneto-resistance, Bose-Einstein condensation, the formation of excitonic gases, or the integer and fractional Quantum Hall effects. The discovery of high-Tc superconductivity in particular was a watershed event, leading to dramatic experimental and theoretical advances in the field of correlated-electron systems. Such materials often exhibit competition between the charge, lattice, spin, and orbital degrees of freedom, whose cause-effect relationships are difficult to ascertain. Experimental insight into the properties of solids is traditionally obtained by time-averaged probes, which measure e.g., linear optical spectra, electrical conduction properties, or the occupied band structure in thermal equilibrium. Many novel physical properties arise from excitations out of the ground state into energetically higher states by thermal, optical, or electrical means. This leads to fundamental interactions between the system's constituents, such as electron-phonon and electron-electron interactions, which occur on ultrafast timescales. While these interactions underlie the physical properties of solids, they are often only indirectly inferred from time-averaged measurements. Time-resolved spectroscopy, consequently, is playing an ever increasing role to provide insight into light-matter interaction, microscopic processes, or cause-effect relationships that determine the physics of complex materials. In the past, experiments using visible and near-infrared femtosecond pulses have been extensively employed, e.g. to follow relaxation and dephasing processes in metals and semiconductors. However, many basic excitations in strongly-correlated electron systems and nanoscale materials occur at lower energies. The terahertz (THz) regime is particularly rich in such fundamental resonances. This includes ubiquitous lattice vibrations and low-energy collective oscillations of conduction charges. In nanoscale materials, band structure quantization also yields novel infrared and THz transitions, including intersubband absorption in quantum wells. The formation of excitons in turn leads to low-energy excitations analogous to inter-level transitions in atoms. In transition-metal oxides, fundamental excitation gaps arise from charge pairing into superconducting condensates and other correlated states. This motivates the use of ultrafast THz spectroscopy as a powerful tool to study light-matter interactions and microscopic processes in nanoscale and correlated-electron materials. A distinct advantage of coherent THz pulses is that the amplitude and phase of the electric field can be measured directly, as the THz fields are coherent with the fs pulses from which they are generated. Using THz time-domain spectroscopy (THz-TDS), both the real and imaginary parts of the response functions (such as the dielectric function) are obtained directly without the need for Kramers?Kronig transforms. The THz response can also be expressed in terms of absorption and refractive index, or as the optical conductivity. The optical conductivity describes the current response of a many-body system to an electric field, an ideal tool to study conducting systems. A second important advantage is the ultrafast time resolution that results from the short temporal duration of the THz time-domain sources. In particular, optical-pump THz-probe spectroscopy enables a delicate probe of the transient THz conductivity after optical photoexcitation. These experiments can provide insight into quasiparticle interactions, phase transitions, or nonequilibrium dynamics. In this chapter we will provide many such examples. Since THz spectroscopy of solids is a quickly expanding field
