Because of the local and global impacts of algae blooms and patchiness on water quality, carbon cycling and climate, models of plankton dynamics are of current interest. Here the temporal and spatial patterns in natural plankton communities are interpreted as transient and stationary non-equilibrium solutions of dynamical nonlinear interaction-diffusion-advection systems. The analysis of a simple model of phytoplankton-zooplankton dynamics in space and time is presented. Nutrients and planktivorous fish are not treated separately, but they act as environmental control variables. After summarizing the local properties, the emergence of spatial and spatio-temporal patterns is considered. Conditions for the formation of spatial patterns after the diffusive instability of a uniform species distribution are derived, and numerical results of diffusive as well as shear-diffusive patterning are presented on a two-dimensional spatial cross-section. The variability of phytoplankton growth due to its light sensitivity is considered in a linear approximation. Multiple stability, oscillations, the propagation of planktic fronts as well as stationary and drifting patches are presented.