With the tremendous successes of the electronics and photonics industries, the performance of common device materials, such as silicon or III-V's, has been pushed nearly to the limit. New materials, with unconventional properties, are required for continued technological progress. Low dimensional and nanostructured materials have unusual electronic, mechanical, plasmonic, and optical properties which are crucial to evaluate for novel device applications. I will present optical and terahertz spectroscopic measurements of the conductivity and carrier dynamics in two material systems which are of interest for two-dimensional electronics and optoelectronics: graphene and molybdenum disulfide. Using time-resolved spectroscopy, we have investigated the time scales and physics of electron cooling and recombination in these materials, which are critical for many device applications. Using frequency-resolved terahertz spectroscopy, we have investigated the nature of their conductivities. In molybdenum disulfide, we have found for the first time the intrinsic mobility at low temperature, which is an order of magnitude larger than previously measured. In graphene microstructures, we have demonstrated that the intraband conductivity is dominated by plasmonic effects, and we have investigated the role of interactions between nearby plasmonic devices.