The terahertz (THz) frequency range. 0.1 - 20 THz, exists between the microwave and infrared ranges and contains abundant information on the dynamics of charge and spin carriers in condensed matter systems. Since its advent two decades ago, THz spectroscopy has been extensively used to study a wide range of solid state materials, including typical semiconductors, conducting polymers, insulators, superconductors, and artificially grown structures such as quantum wells. In these systems, electronic and photonic events tend to occur on the time scale of tens to hundreds of femtoseconds, which results in many important excitations, resonances and dynamical phenomena in the THz frequency range. In this dissertation work, we have developed a typical THz time-domain spectroscopy (TDS) system to investigate the THz dynamics of quantum-confined electrons in two important types of carbon nanomaterial: single-walled carbon nanotubes (SWNTs) and graphene. Polarization dependent THz transmission measurements were conducted on a highly-aligned SWNT film on a sapphire substrate, revealing extremely high anisotropy: virtually no attenuation was observed when the polarization of the THz beam was perpendicular to the nanotube axis, while the THz beam was strongly absorbed when its polarization was parallel to the tube axis. From the measured absorption anisotropy, we calculated the reduced linear dichrosim to be 3, corresponding to a nematic order parameter of 1. These observations are a direct result of the one-dimensional nature of conduction electrons in the nanotubes and at the same time, demonstrate that any misalignment of nanotubes in the film mast have characteristic length scales much smaller than the wavelengths used in these experiments (1.5 mm - 150 μm). Based on this work, an ideal THz linear polarizer built with parallel stacks of such aligned SWNT films was synthesized, exhibiting a degree of polarization of 99.9% throughout the frequency range 0.2 - 2.2 THz and a high extinction ratio of 10 -3 (or 30 dB). The THz complex conductivity of the thin SWNT film was extracted through a proper model directly from the TDS data without Kramers-Kronig analysis. Both real and imaginary parts of the conductivity showed a non-Drude frequency dependence, indicating the presence of plasmon-dipole resonance at higher frequencies. Finally, the optical conductivity of large-area. graphene grown from solid state carbon source was studied in a wide spectral range (7 cm -1 - 9500 cm -1 ) using THz-TDS and Fourier transform infrared spectroscopy. We observed that the Fermi level E f of graphene could be tuned by both electrical gating and thermal annealing. The optical conductivity measured at different carrier concentrations exhibited Drude-like frequency dependence, and different 2 E f onsets in the spectrum were probed as well.