Geometric properties of Lie groups with left invariant Riemannian metric have been studied extensively. For example, the Milnor's classication of 3-dimensional Lie groups with left invariant Riemannian metric is classical reference. The geometry of Lie groups with left invariant pseudo-Riemannian metric is not so well known and with many open questions. Motivated by previously mentioned facts and our discussions with V. Matveev we investigate Lorentz geometry of 4-dimensional nilpotent Lie groups. First, we classify left invariant Lorentz metrics on 4-dimensional nilpotent Lie groups H3xR and G4. This research is motivated by results of Lauret where 3 and 4-dimensional Riemannian nilmanifolds are classified. As expected, we found much more metrics than in the Riemannian case. Note that Cordero and Parker
solved analogue problem for Lorentz 3-dimensional Lie groups. The geometry of these metrics is investigated. We calculate curvature tensor and holonomy groups of the metrics and investigate decomposability of the metrics. It is interesting that some of metrics have parallel null vector, but are not decomposable. These are exactly pp-wave metrics. Finally, we find projective classes of the metrics. We reveal that some of
metrics are geometrically rigid, while others have projectively equivalent metrics that are also affinely equivalent. They are either decomposable metrics or indecomposable metrics with parallel null vector. It is interesting that the affinely equivalent metrics are also left invariant. We also check that in the Riemannian case (metrics from Lauret classication) all indecomposable metrics are geodesically rigid.