The notion of a harmonic Z/2 spinor was introduced by Taubes as an abstraction of various limiting objects appearing in compactifications of gauge-theoretic moduli spaces. I will explain this notion and discuss an existence result for harmonic Z/2 spinors on three-manifolds. The proof uses a wall-crossing formula for solutions to generalized Seiberg-Witten equations in dimension three, a result itself motivated by Yang-Mills theory on manifolds with exceptional holonomy. The talk is based on joint work with Thomas Walpuski.