  A general class of Lorentzian metrics, $M_0 x R^2$, $ds^2 = <.,.> + 2 du dv +
H(x,u) du^2$, with $(M_0, <.,.>$ any Riemannian manifold, is introduced in
order to generalize classical exact plane fronted waves. Here, we start a
systematic study of their main geodesic properties: geodesic completeness,
geodesic connectedness and multiplicity, causal character of connecting
geodesics. These results are independent of the possibility of a full
integration of geodesic equations. Variational and geometrical techniques are
applied systematically.
  In particular, we prove that the asymptotic behavior of $H(x,u)$ with $x$ at
infinity determines many properties of geodesics. Essentially, a subquadratic
growth of $H$ ensures geodesic completeness and connectedness, while the
critical situation appears when $H(x,u)$ behaves in some direction as $|x|^2$,
as in the classical model of exact gravitational waves
