Species patchiness implies that nearby observations of species abundance tend to be similar or that individual conspecific organisms are more closely spaced than by random chance. This can be caused either by the positive spatial autocorrelation among the locations of individual organisms due to ecological spatial processes (e.g., species dispersal, competition for space and resources) or by spatial dependence due to (positive or negative) species responses to underlying environmental conditions. Both forms of spatial structure pose problems for statistical analysis, as spatial autocorrelation in the residuals violates the assumption of independent observations, while environmental heterogeneity restricts the comparability of replicates. In this paper, we discuss how spatial structure due to ecological spatial processes and spatial dependence affects spatial statistics, landscape metrics, and statistical modeling of the species-environment correlation. For instance, while spatial statistics can quantify spatial pattern due to an endogeneous spatial process, these methods are severely affected by landscape environmental heterogeneity. Therefore, statistical models of species response to the environment not only need to accommodate spatial structure, but need to distinguish between components due to exogeneous and endogeneous processes rather than discarding all spatial variance. To discriminate between different components of spatial structure, we suggest using (multivariate) spatial analysis of residuals or delineating the spatial realms of a stationary spatial process using boundary detection algorithms. We end by identifying conceptual and statistical challenges that need to be addressed for adequate spatial analysis of landscapes.