The human distal gut harbours a vast ensemble of microbes (the microbiota) that provide important metabolic capabilities, including the ability to extract energy from otherwise indigestible dietary polysaccharides. Studies of a few unrelated, healthy adults have revealed substantial diversity in their gut communities, as measured by sequencing 16S rRNA genes, yet how this diversity relates to function and to the rest of the genes in the collective genomes of the microbiota (the gut microbiome) remains obscure. Studies of lean and obese mice suggest that the gut microbiota affects energy balance by influencing the efficiency of calorie harvest from the diet, and how this harvested energy is used and stored. Here we characterize the faecal microbial communities of adult female monozygotic and dizygotic twin pairs concordant for leanness or obesity, and their mothers, to address how host genotype, environmental exposure and host adiposity influence the gut microbiome. Analysis of 154 individuals yielded 9,920 near full-length and 1,937,461 partial bacterial 16S rRNA sequences, plus 2.14 gigabases from their microbiomes. The results reveal that the human gut microbiome is shared among family members, but that each person's gut microbial community varies in the specific bacterial lineages present, with a comparable degree of co-variation between adult monozygotic and dizygotic twin pairs. However, there was a wide array of shared microbial genes among sampled individuals, comprising an extensive, identifiable `core microbiome' at the gene, rather than at the organismal lineage, level. Obesity is associated with phylum-level changes in the microbiota, reduced bacterial diversity and altered representation of bacterial genes and metabolic pathways. These results demonstrate that a diversity of organismal assemblages can nonetheless yield a core microbiome at a functional level, and that deviations from this core are associated with different physiological states (obese compared with lean).