  Drawing parallels with hyperplane arrangements, we develop the theory of
arrangements of submanifolds. Given a smooth, finite dimensional, real manifold
$X$ we consider a finite collection $\mathcal{A}$ of locally flat,
codimension-1 submanifolds that intersect like hyperplanes. To such a
collection we associate two combinatorial objects: the face category and the
intersection poset. We also associate a topological space to the arrangement
called the tangent bundle complement. It is the complement of union of tangent
bundles of these submanifolds inside the tangent bundle of the ambient
manifold. Our aim is to investigate the relationship between the combinatorics
of the arrangement and the topology of the complement. In particular we show
that the tangent bundle complement has the homotopy type of a finite cell
complex. We generalize the classical theorem of Salvetti for hyperplane
arrangements and show that this particular cell complex is completely
determined by the face category.
