  In this paper, we discuss the conceptual underpinnings of Modern Coexistence
Theory (MCT), a quantitative framework for understanding ecological
coexistence. In order to use MCT to infer how species are coexisting, one must
relate a complex model (which simulates coexistence in the real world) to
simple models in which previously proposed explanations for coexistence have
been codified. This can be accomplished in three steps: 1) relating the
construct of coexistence to invasion growth rates, 2) mathematically
partitioning the invasion growth rates into coexistence mechanisms (i.e.,
classes of explanations for coexistence), and 3) relating coexistence
mechanisms to simple explanations for coexistence. Previous research has
primarily focused on step 2. Here, we discuss the other crucial steps and their
implications for inferring the mechanisms of coexistence in real communities.
  Our discussion of step 3 -- relating coexistence mechanisms to simple
explanations for coexistence -- serves a heuristic guide for hypothesizing
about the causes of coexistence in new models; but also addresses
misconceptions about coexistence mechanisms. For example, the storage effect
has little to do with bet-hedging or "storage" via a robust life-history stage;
relative nonlinearity is more likely to promote coexistence than originally
thought; and fitness-density covariance is an amalgam of a large number of
previously proposed explanations for coexistence (e.g., the
competition-colonization trade-off, heteromyopia, spatially-varying resource
supply ratios). Additionally, we review a number of topics in MCT, including
the role of "scaling factors"; whether coexistence mechanisms are
approximations; whether the magnitude or sign of invasion growth rates matters
more; whether Hutchinson solved the paradox of the plankton; the
scale-dependence of coexistence mechanisms; and much more.
