Light beams carrying orbital angular momentum (OAM) have become a mainstay of optical science and technology. In these beams, well-known examples of which are the Laguerre-Gaussian (LG_pm ) and Bessel-Gaussian (BG_m ) beams, the OAM vector points parallel or anti-parallel to propagation, and is associated with a phase winding 2πm in the plane transverse to the propagation direction, where integer m is the winding order or the “topological charge”. Such beams can be monochromatic.Recently, our group discovered a new type of OAM structure that naturally emerges from nonlinear self-focusing, which we dubbed the spatio-temporal optical vortex (STOV). Here, the phase winding exists in a spatiotemporal plane, with the OAM pointing transverse to propagation. In this dissertation, we extend the generation of STOV-carrying pulses to the linear regime, demonstrating their generation using a 4f pulse shaper and measuring their free-space propagation using a new ultrafast single-shot space- and time-resolving diagnostic, TG-SSSI (transient-grating single-shot supercontinuum spectral interferometry). We then demonstrate that transverse OAM is a property of photons by experimentally confirming the conservation of transverse OAM in second harmonic generation. Because the field of STOVs is so new, a first principles theory for their transverse OAM was lacking. We developed such a theory for transverse OAM that predicts half integer values of OAM and the existence of a STOV polariton in dispersive media. The surprise of half-integer OAM values launched a debate in the OAM community, which has been resolved in favor of our theory by our most recent experiments. These explore how phase and amplitude perturbations can impart spatiotemporal torques to light. We find that transverse OAM can be imparted to light pulses only for (1) sufficiently fast transient phase perturbations or (2) energy removal from a pulse already possessing transverse OAM.