  Ecosystems display a complex spatial organization. Ecologists have long tried
to characterize them by looking at how different measures of biodiversity
change across spatial scales. Ecological neutral theory has provided simple
predictions accounting for general empirical patterns in communities of
competing species. However, while neutral theory in well-mixed ecosystems is
mathematically well understood, spatial models still present several open
problems, limiting the quantitative understanding of spatial biodiversity. In
this review, we discuss the state of the art in spatial neutral theory. We
emphasize the connection between spatial ecological models and the physics of
non-equilibrium phase transitions and how concepts developed in statistical
physics translate in population dynamics, and vice versa. We focus on
non-trivial scaling laws arising at the critical dimension D = 2 of spatial
neutral models, and their relevance for biological populations inhabiting
two-dimensional environments. We conclude by discussing models incorporating
non-neutral effects in the form of spatial and temporal disorder, and analyze
how their predictions deviate from those of purely neutral theories.
