  Our primary aim is to develop a theory of equivariant genera for stably
complex manifolds equipped with compatible actions of a torus T^k. In the case
of omnioriented quasitoric manifolds, we present computations that depend only
on their defining combinatorial data; these draw inspiration from analogous
calculations in toric geometry, which seek to express arithmetic, elliptic, and
associated genera of toric varieties in terms only of their fans. Our theory
focuses on the universal toric genus \Phi, which was introduced independently
by Krichever and Loeffler in 1974, albeit from radically different viewpoints.
In fact \Phi is a version of tom Dieck's bundling transformation of 1970,
defined on T^k-equivariant complex cobordism classes and taking values in the
complex cobordism algebra of the classifying space. We proceed by combining the
analytic, the formal group theoretic, and the homotopical approaches to genera,
and refer to the index theoretic approach as a recurring source of insight and
motivation. The resultant flexibility allows us to identify several distinct
genera within our framework, and to introduce parametrised versions that apply
to bundles equipped with a stably complex structure on the tangents along their
fibres. In the presence of isolated fixed points, we obtain universal
localisation formulae, whose applications include the identification of
Krichever's generalised elliptic genus as universal amongst genera that are
rigid on SU-manifolds. We follow the traditions of toric geometry by working
with a variety of illustrative examples wherever possible. For background and
prerequisites we attempt to reconcile the literature of east and west, which
developed independently for several decades after the 1960s.
