First demonstrated by Isaac Newton in his prism experiment, angular dispersion (AD) is a prevalent effect in optics where each wavelength in a pulsed field propagates at a different angle. Angular dispersion occurs after a collimated pulse traverses a diffractive or dispersive device and, as a result, helps modify the group velocity of a pulse in free space and introduces group-velocity dispersion into the freely propagating wave packet. These are essential ingredients in group-velocity matching and dispersion cancellation in various optical settings. With 300 years of development, it was only recently that a new class of angular dispersion materialized as non-differentiable AD. This non-differentiable AD has also been studied under the moniker space-time wave packets (STWP) and has shown to be propagation-invariant and possess arbitrary group velocity. In this dissertation, I will study (1) the underpinning theory of how non-differentiable AD allows for an optical field to break the pre-conceived notions of group velocity, group velocity dispersion (GVD), and pulse front tilt for on-axis propagation through analytical and experimental demonstrations. From these developments, I will (2) apply these concepts of non-differentiable AD to dispersive materials. I will validate these analytical predictions through experiments showing that propagation-invariant wave packets can also be supported in normal and anomalous media. Moreover, I will prove, through the use of non-differentiable AD, that the dispersive properties of a material can be overwritten to produce arbitrary group velocity and GVD characteristics. With this new information on propagation-invariant fields in dispersive materials, I will (3) exhibit new classes of optical fields that were previously theorized but never synthesized in dispersive materials, such as the X- to O- transition in anomalous GVD materials, which will be connected to the de-Broglie-Mackinnon wave packet and particle wave packets. To address the propagation invariance of non-differentiable AD, I will (4) demonstrate the STWP propagation throughout a kilometer in a turbulent environment and develop a new Rayleigh length for the STWP. Finally, I will (5) establish the consequences of discretization on the non-differentiable AD and produce a new form of the Talbot effect in which the temporal and spatial degrees of freedom are interlocked along with independent spatial and temporal Talbot effects in free space.