Our aim is to develop topological analogues of an ongoing programme in toric
geometry, which seeks to express arithmetic, elliptic, and related genera of
toric varieties as functions of their fans. In this context, we introduce
methods for computing equivariant genera of omnioriented quasitoric manifolds M
purely in terms of the combinatorial data (P,\Lambda) by which such M are
determined. We develop the theory around the universal example \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, being defined on T^k-equivariant complex
cobordism classes and taking values in the complex cobordism algebra
\Omega^*_U(BT^k_+) of the classifying space of the k-torus. We proceed by
combining three approaches to genera, namely the analytic, the formal group
theoretic, and the homotopical. These provide the freedom to study 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 case of isolated fixed points we obtain universal
localisation formulae; their applications include the identification of
Krichever's generalised elliptic genus as universal amongst genera that are
rigid on SU-manifolds. For background and prerequisites we draw on the
literature of both east and west, which developed independently for several
decades following the mid 1960s.