Far from equilibrium, comparatively little is known about the possibilities nature reserves for the structure and states of quantum many-body systems. A potential scenario is that these systems can approach a non-thermal fixed point and show universal scaling dynamics. The associated spatio-temporal self-similar evolution of correlations is characterized by universal scaling functions and scaling exponents. In this thesis, we investigate the universal scaling behavior of multi-component bosonic quantum gases from a theoretical point of view. In particular, we perform numerical simulations of spin-1 Bose gases in one and two spatial dimensions. To enable universal scaling dynamics, we prepare far-from-equilibrium initial configurations by making use of instabilities arising from a parameter quench between different phases of the spin-1 model. The subsequent universal scaling at the non-thermal fixed point is driven by the annihilation and dissolution of (quasi)topological excitations. In addition, we make analytical predictions for the non-thermal fixed point scaling of U (N)-symmetric models which we corroborate with numerical simulations of a U (3)-symmetric Bose gas in three spatial dimensions. We find that the scaling behavior at the fixed point is dominated by the conserved redistribution of collective excitations. Furthermore, we introduce prescaling as a generic feature of the evolution of a quantum many-body system towards a non-thermal fixed point. During the prescaling evolution, some well-measurable properties of spatial correlations already scale with the universal exponents of the fixed point while others still show scaling violations. We illustrate the existence of prescaling by means of numerical simulations of a three-dimensional U (3)-symmetric Bose gas. The research presented in this thesis contributes to a deeper understanding of universal scaling dynamics far from equilibrium. In particular, it unravels important key aspects for establishing out-of- equilibrium universality classes. Furthermore, the introduced concept of prescaling allows bridging the gap in the time evolution from the initial state to the associated non-thermal fixed point.