In this dissertation, we introduce and explore several new optical methods that use the orbital and spin angular momentum of light to probe a variety of solids. We first introduce a new type of pump-probe spectroscopy, in which Laguerre-Gaussian (LG) optical pump pulses impart orbital angular momentum to the electronic states of a material, and the induced orbital angular dichroism is tracked with 100 fs time resolution. Proof of concept measurements on bulk GaAs yield transients that evolve on time scales different from population and spin relaxation as expected, but with surprisingly large lifetimes. We also carry out simulation work on the generation and propagation of the Laguerre-Gaussian modes used as pump pulses in this experiment. We find the spatial distribution of the orbital angular momentum over a cross section of the beam normal to its direction of propagation is not constant, but varies over the pulse duration. We then introduce two more spectroscopic methods for probing the interaction of LG beams with matter. In the first, we consider the differential reflection of opposite handed LG mode components of a Gaussian beam by the Abrikosov flux lattice of a Type-II superconductor. We conduct proof of principle measurements on the Type-II superconductor $\text{Mo}{.79}\text{Ge}{.21}$ but fail to find signal. A simple calculation suggests using a smaller probe spot size would yield a large increase in signal. In the second, we look to see whether there is an orbital parallel to the phenomenon by which cholesteric liquid crystals selectively reflect light that is circularly polarized in a sense identical to their helicity. We pursue one of two proposed experimental tests, and find evidence of non-trivial interactions between OAM light and cholseteric liquid crystals. Next, by demonstrating that time-resolved Kerr rotation (TRKR) measurements have essentially identical temporal profiles when the probe energy is tuned far below, far above, and on resonance with a semiconducting band gap, we show that off-resonance TRKR is a quantitatively accurate tool to measure spin dynamics. We then take off-resonance TRKR measurements on the gapless Weyl semi-metal Mo${.9}$W${.1}$Te$_2$ and find a transient which persists over several hundred picoseconds and does not precess or dephase in the presence of even relatively large transverse applied fields. We attribute this behavior to the spin polarized pockets of the band structure located roughly at the Fermi-energy of the material. We close by calculating the energy dependence of Kerr rotation around a single resonance. We find that it has an even Lorenztian energy dependence, despite often being assumed to mimic the form of Faraday rotation, which has an odd Lorentzian dependence. We take TRKR measurements as a function of probe energy on GaAs and find it has an odd Lorentzian form, in conflict with our derived theoretical result. We show that this discrepancy is resolved by accounting for a decrease in the transition linewidth as the optical energy increases above the band-edge transition.