  The simplest theories often have much merit and many limitations, and in this
vein, the value of Neutral Theory (NT) has been the subject of much debate over
the past 15 years. NT was proposed at the turn of the century by Stephen
Hubbell to explain pervasive patterns observed in the organization of
ecosystems. Its originally tepid reception among ecologists contrasted starkly
with the excitement it caused among physicists and mathematicians. Indeed, NT
spawned several theoretical studies that attempted to explain empirical data
and predicted trends of quantities that had not yet been studied. While there
are a few reviews of NT oriented towards ecologists, our goal here is to review
the quantitative results of NT and its extensions for physicists who are
interested in learning what NT is, what its successes are and what important
problems remain unresolved. Furthermore, we hope that this review could also be
of interest to theoretical ecologists because many potentially interesting
results are buried in the vast NT literature. We propose to make these more
accessible by extracting them and presenting them in a logical fashion. We
conclude the review by discussing how one might introduce realistic non-neutral
elements into the current models.
