Less than two years ago some physicists gave the official announcement that gravitational waves exist, but, from a geometrical point of view, they have always been “real objects” and their properties have been widely investigated.
The aim of this talk is introducing generalized plane waves and discussing some of their properties such as geodesic connectedness and geodesic completeness.

In the last years a great deal of e ort has been invested in order to investigate the existence of geodesics connecting two given points in Lorentzian manifolds as a statement similar to the Hopf-Rinow Theorem is not known for spacetimes.
In particular, some geometric sufficient conditions have been introduced so that stationary spacetimes, i.e. spacetimes equipped with a timelike Killing vector eld, are geodesically connected. Unluckily, a similar global result is not true for space- times equipped with a lightlike Killing vector eld. Anyway, points connected by geodesics can be characterized.
Joint works with Rossella Bartolo and José Luis Flores.