We explain how species accumulation curves are influenced by species richness (total number of species), relative abundance and diversity using computer-generated simulations. Species richness defines the boundary of the horizontal asymptote value for a species accumulation curve, and the shape of the curve is influenced by both relative abundance and diversity. Simulations with a high proportion of rare species and a few abundant species have a species accumulation curve with a low 'shoulder' (inflection point on the ordinate axis) and a long upward slope to the asymptote. Simulations with a high proportion of relatively abundant species have a steeply rising initial slope to the species accumulation curve and plateau early. Diversity (as measured by Simpson's and Shannon-Weaver indices) for simulations is positively correlated with the initial slope of the species accumulation curve. Species accumulation curves cross when one simulation has a high proportion of both rare and abundant species compared with another that has a more even distribution of abundance among species.