BPS states in N=2 gauge theories or string vacua are generically stable but liable to decay across certain codimension-one loci in moduli space. This process is easily understood by viewing BPS states as a bound state of more elementary BPS constituents, described classically by multi-centered solutions of the low energy effective action. The semi-classical quantization of the space of such solutions agrees with the wall-crossing formulae derived in the mathematical literature on BPS invariants, providing a physically elementary justification of the latter. Using this intuition, it is possible to express the BPS index, at any point in moduli space, in terms of indices associated to elementary (or single-centered) constituents. If time permits, I will present evidence for this idea in the case of BPS states described by quiver representations.