Spatially extended population dynamics models that incorporate intrinsic noise serve as case studies for the role of fluctuations and correlations in biological systems. Including spatial structure and stochastic noise in predator-prey competition invalidates the deterministic Lotka-Volterra picture of neutral population cycles. Stochastic models yield long-lived erratic population oscillations stemming from a resonant amplification mechanism. In spatially extended predator-prey systems, one observes noise-stabilized activity and persistent correlations. Fluctuation-induced renormalizations of the oscillation parameters can be analyzed perturbatively. The critical dynamics and the non-equilibrium relaxation kinetics at the predator extinction threshold are characterized by the directed percolation universality class. Spatial or environmental variability results in more localized patches which enhances both species densities. Affixing variable rates to individual particles and allowing for trait inheritance subject to mutations induces fast evolutionary dynamics for the rate distributions. Stochastic spatial variants of cyclic competition with rock-paper-scissors interactions illustrate connections between population dynamics and evolutionary game theory, and demonstrate how space can help maintain diversity. In two dimensions, three-species cyclic competition models of the May-Leonard type are characterized by the emergence of spiral patterns whose properties are elucidated by a mapping onto a complex Ginzburg-Landau equation. Extensions to general food networks can be classified on the mean-field level, which provides both a fundamental understanding of ensuing cooperativity and emergence of alliances. Novel space-time patterns emerge as a result of the formation of competing alliances, such as coarsening domains that each incorporate rock-paper-scissors competition games.