For principled and substantially philosophical reasons, based largely on his reform of natural history by inverting the Paleyan notion of overarching and purposeful beneficence in the construction of organisms, Darwin built his theory of selection at the single causal level of individual bodies engaged in unconscious (and metaphorical) struggle for their own reproductive success. But the central logic of the theory allows selection to work effectively on entities at several levels of a genealogical hierarchy, provided that they embody a set of requisite features for defining evolutionary individuality. Genes, cell lineages, demes, species, and clades, as well as Darwin's favoured organisms, embody these requisite features in enough cases to form important levels of selection in the history of life. R. A. Fisher explicitly recognized the unassailable logic of species selection, but denied that this real process could be important in evolution because, compared with the production of new organisms within a species, the origin of new species is so rare, and the number of species within most clades so low. I review this and other classical arguments against higher–level selection, and conclude (in the first part of this paper) that they are invalid in practice for interdemic selection, and false in principle for species selection. Punctuated equilibrium defines the individuality of species and refutes Fisher's classical argument based on cycle time. In the second part of the paper, I argue that we have failed to appreciate the range and power of selection at levels above and below the organismic because we falsely extrapolate the defining properties of organisms to these other levels (which are characterized by quite different distinctive features), and then regard the other levels as impotent because their effective individuals differ so much from organisms. We would better appreciate the power and generality of hierarchical models of selection if we grasped two key principles: first, that levels can interact in all modes (positively, negatively, and orthogonally), and not only in the negative style (with a higher level suppressing an opposing force of selection from the lower level) that, for heuristic and operational reasons, has received almost exclusive attention in the existing literature; and second, that each hierarchical level differs from all others in substantial and interesting ways, both in the style and frequency of patterns in change and causal modes.