In this dissertation, I study a nonlinear partial differential equation, the complex Ginzburg-Landau (CGL) equation. I first employed the perturbative field-theoretic renormalization group method to investigate the critical dynamics near the continuous non-equilibrium transition limit in this equation with additive noise. Due to the fact that time translation invariance is broken following a critical quench from a random initial configuration, an independent ``initial-slip'' exponent emerges to describe the crossover temporal window between microscopic time scales and the asymptotic long-time regime. My analytic work shows that to first order in a dimensional expansion with respect to the upper critical dimension, the extracted initial-slip exponent in the complex Ginzburg-Landau equation is identical to that of the equilibrium model A. Subsequently, I studied transient behavior in the CGL through numerical calculations. I developed my own code to numerically solve this partial differential equation on a two-dimensional square lattice with periodic boundary conditions, subject to random initial configurations. Aging phenomena are demonstrated in systems with either focusing and defocusing spiral waves, and the related aging exponents, as well as the auto-correlation exponents, are numerically determined. I also investigated nucleation processes when the system is transiting from a turbulent state to the ``frozen'' state. An extracted finite dimensionless barrier in the deep-quenched case and the exponentially decaying distribution of the nucleation times in the near-transition limit are both suggestive that the dynamical transition observed here is discontinuous. This research is supported by the U. S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award DE-FG02-SC0002308 / Doctor of Philosophy / The complex Ginzburg-Landau equation is one of the most studied nonlinear partial differential equation in the physics community. I study this equation using both analytical and numerical methods. First, I employed the field theory approach to extract the critical initial-slip exponent, which emerges due to the breaking of time translation symmetry and describes the intermediate temporal window between microscopic time scales and the asymptotic long-time regime. I also numerically solved this equation on a two-dimensional square lattice. I studied the scaling behavior in non-equilibrium relaxation processes in situations where defects are interactive but not subject to strong fluctuations. I observed nucleation processes when the system under goes a transition from a strongly fluctuating disordered state to the relatively stable “frozen” state where its dynamics cease. I extracted a finite dimensionless barrier for systems that are quenched deep into the frozen state regime. An exponentially decaying long tail in the nucleation time distribution is found, which suggests a discontinuous transition. This research is supported by the U. S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award DE-FG02-SC0002308.