The study of the electronic band structure of solids is a central task in materials science. Indeed, knowing a material's band structure is key to an understanding of its macroscopic electronic properties. In recent years, a new class of materials has emerged whose unifying feature is an electronic structure that breaks with conventional band theory. Rather than obeying the usual Schrödinger Hamiltonian, the low-energy quasiparticle excitations in the band structure of these materials are better described as chiral, massless Dirac-like carriers, similar in nature to the relativistic quantum particles known from high-energy physics. Prominent members of this class are graphene and topological insulators. Despite being vastly different materials, they share a number of intriguing electronic properties that render them highly attractive for both fundamental and application-oriented studies. The thrust of this thesis work is to investigate the photoexcited Dirac carriers in graphene and in the series of topological insulators (Sb2)m-Sb2Te3 by means of angle-resolved photoemission spectroscopy (ARPES) and XUV laser-based time-resolved ARPES. The aim is to push the research frontier further towards a more complete understanding of the fate of a Dirac carrier when it is brought out of thermodynamic equilibrium by an optical excitation; and what role the substrate material, doping level, laser excitation and the introduction of a band gap play, as well as how one can possibly engineer such a Dirac-like electronic structure. To this end, we take advantage of ARPES' momentum and energy resolving capabilities that can be extended into the time domain by adding femtosecond time-resolution in a pump-probe scheme. By taking the Dirac carriers out of their equilibrium state by a laser pump pulse and watching directly with a second probe pulse how they interact and relax back to equilibrium, a movie of the in situ carrier dynamics is obtained. This dynamical view on the ultrafast scattering processes of photoexcited Dirac carriers in graphene allows us to show a quasi-instant sub 40 femtosecond thermalization to a hot carrier distribution. The carriers cool back to equilibrium on femto- and picosecond timescales via excitations of optical phonons and acoustic phonons assisted by disorder-scattering in supercollisions processes. A method is introduced to accurately extract the hot carriers¿ temperature that is used to quantify the carrier multiplication. Cooling times and carrier multiplication are found to display a remarkable dependence on the doping level of graphene to the extent that a substantial carrier multiplication is possible. The hot carrier dynamics is found to be severely perturbed by the presence of a metal, and also pump-induced electric fields are shown to affect a correct interpretation of TR-ARPES data by masking the intrinsic carrier dynamics. Furthermore, we demonstrate that, in the case of bilayer graphene, the existence of a single-particle band gap in the electronic structure leads to a markedly enhanced carrier lifetime over monolayer graphene, despite the existence of in-gap states. Finally, a complete picture of the evolution of the band structure and Dirac nature of the carriers across a series of compounds, from Sb2Te3 to elemental Sb, is drawn, suggesting a promising strategy for engineering the electronic states.