We study transversely Lorentzian foliations of closed 3-manifolds. We classify them under a completeness hypothesis and deduce the dual classification of codimension-one geodesically complete timelike totally geodesic foliations. Also we provide an example of a Lorentzian foliation of a compact 3-manifold which is equicontinuous on a proper dense open subset of the manifold, and hence not transversely complete.