  Feynman diagrams (notably the triangle diagram) involving heavy enough
particles contain branch cuts on the physical sheet - anomalous thresholds -
which, unlike normal thresholds and bound-state poles, do not correspond to any
asymptotic $n$-particle state. ``Who ordered that?" We show that anomalous
thresholds arise as a consequence of established S-matrix principles and two
reasonable assumptions: unitarity below the physical region and analyticity in
the mass. We find explicit nonperturbative formulas for the discontinuity
across the anomalous threshold in $d=2$, and in $d = 4$, ready to be used in
dispersion relations for bootstrap and phenomenological applications.
