For an arbitrary meet-semilattice we introduce notions of combinatorial
blowups, building sets, and nested sets. This gives a common abstract
framework for the incidence combinatorics occurring in the context of
DeConcini-Procesi models of subspace arrangements and resolutions
of singularities in toric varieties.
Our main theorem states that a sequence of combinatorial blowups,
prescribed by a building set in linear extension compatible order,
gives the face poset of the simplicial complex of nested sets.
As applications we trace the incidence combinatorics through every
step of the DeConcini-Procesi model construction, and we introduce
the notions of building sets and nested sets to the context of toric