An important phenomenon of systems driven far-from-equilibrium is spatiotemporal chaos where the dynamics are aperiodic in both time and space. We explored this numerically for three systems: the Lorenz-96 model, the Swift-Hohenberg equation, and Rayleigh-Bénard convection. The Lorenz-96 model is a continuous in time and discrete in space phenomenological model that captures important features of atmosphere dynamics. We computed the fractal dimension as a function of system size and external forcing to estimate characteristic length and time scales describing the chaotic dynamics. We found extensive chaos with significant deviations from extensivity for small changes in system size and also the power-law growth of the dimension with increasing forcing. The Swift-Hohenberg equation is a partial differential equation for a scalar field, which has been widely used as a model for the study of pattern formation. We found that the magnitude of the mean flow in this model must be sufficiently large for spiral defect chaos to occur. We also explored the spatiotemporal chaos in experimentally accessible Rayleigh-Bénard convection using large-scale numerical simulations of the Boussinesq equations and the corresponding tangent space equations. We performed a careful study analyzing the impact of variations in the domain size, Rayleigh number, and Prandtl number on the system dynamics and fractal dimension. In addition, we quantified the dynamics of the spectrum of Lyapunov exponents and the leading order Lyapunov vector in an effort to connect directly with the dynamics of the flow field patterns. Further, we numerically studied the synchronization of chaos in convective flows by imposing time-dependent boundary conditions from a principal domain onto an initially quiescent target domain. We identified a synchronization length scale to quantify the size of a chaotic element using only information from the pattern dynamics. We also explored the relationship of this length scale with the pattern wavelength. Finally, we analyzed bioconvection which occurs as the result of the collective behavior of a suspension of swimming microorganisms. We developed a series of simulations to capture the gyrotactic pattern formation of the swimming algae. The results can be compared with the corresponding trend of pattern instabilities observed in the experimental studies. / Ph. D.