  We show that non-perturbative effects are logarithmically enhanced for
transverse-momentum-dependent observables such as q_T-spectra of electroweak
bosons in hadronic collisions and jet broadening at e^+e^- colliders. This
enhancement arises from the collinear anomaly, a mechanism characteristic for
transverse observables, which induces logarithmic dependence on the hard scale
in the product of the soft and collinear matrix elements. Our analysis is based
on an operator product expansion and provides, for the first time, a
systematic, model-independent way to study non-perturbative effects for this
class of observables. For the case of jet broadening, we relate the leading
correction to the non-perturbative shift of the thrust distribution.
