The tight span, or injective envelope, is one of the most elegant and useful
constructions in metric geometry. Here we introduce a generalisation of
metrics, called diversities, and demonstrate that the rich theory associated to
metric tight spans extends to a seemingly richer theory of diversity tight
spans. Diversities are a variant of metrics that assign values not just to
pairs of elements but to sets of elements. They satisfy a triangle inequality,
and vanish on singletons. We establish basic properties of diversities and
their tight spans, and prove that many of the important properties of, and
results about, metric tight spans extend to the diversity case in a non-trivial
way.