Handedness and cerebral asymmetry are commonly assumed to be uniquely human, and even defining characteristics of our species. This is increasingly refuted by the evidence of behavioural asymmetries in non-human species. Although complex manual skill and language are indeed unique to our species and are represented asymmetrically in the brain, some non-human asymmetries appear to be precursors, and others are shared between humans and non-humans. In all behavioural and cerebral asymmetries so far investigated, a minority of individuals reverse or negate the dominant asymmetry, suggesting that such asymmetries are best understood in the context of the overriding bilateral symmetry of the brain and body, and a trade-off between the relative advantages and disadvantages of symmetry and asymmetry. Genetic models of handedness, for example, typically postulate a gene with two alleles, one disposing towards right-handedness and the other imposing no directional influence. There is as yet no convincing evidence as to the location of this putative gene, suggesting that several genes may be involved, or that the gene may be monomorphic with variations due to environmental or epigenetic influences. Nevertheless, it is suggested that, in behavioural, neurological and evolutionary terms, it may be more profitable to examine the degree rather than the direction of asymmetry.