Various ecological mechanisms influence the forms of species richness relationships (SRRs). These mechanisms can be gathered under five general categories: more individuals, environmental heterogeneity, dispersal limitations, biotic interactions, and multiple species pools. Often only the first two categories are discussed. In contrast, we examine all five and explore how they can influence the form of SRRs. We discuss how various sampling schemes and methods of SRR construction can be used to gain insight about how various processes influence species richness patterns. The field is ripe for probing these effects through more complex simulation models or more sophisticated mathematical approaches. To facilitate deeper understanding, we need to embrace the full spectrum of SRRs and reconsider the assumed common knowledge about the functional form of SRRs. The relationship between species richness and the space or time over which it is sampled has received increasing attention over the past decade, resulting in extensive debates about terminology and methods of construction. These debates reflect deep conceptual issues; to resolve them we discuss the long history of species richness relationships (SRRs) and the connections among different methodological and terminological approaches. We reinforce recent calls to organize the variety of methods used to construct SRRs into a cohesive structure. SRRs are descriptors of various aspects of inventory (a- and c-) diversity and the various types of SRRs serve different purposes. Contrary to most claims, SRRs do not provide a direct measure of differentiation (b-) diversity.