  Starting from the classical notion of an oriented congruence (i.e. a
foliation by oriented curves) in $R^3$, we abstract the notion of an oriented
congruence structure. This is a 3-dimensional CR manifold $(M,H, J)$ with a
preferred splitting of the tangent space $TM=V\oplus H$. We find all local
invariants of such structures using Cartan's equivalence method refining
Cartan's classification of 3-dimensional CR structures. We use these invariants
and perform Fefferman like constructions, to obtain interesting Lorentzian
metrics in four dimensions, which include explicit Ricci-flat and Einstein
metrics, as well as not conformally Einstein Bach-flat metrics.
