Let M = M0 × R 2 be a pp–wave type spacetime endowed with the metric 〈·, ·〉z = 〈·, ·〉x + 2 du dv + H(x, u) du , where (M0, 〈·, ·〉x) is any Riemannian manifold and H(x, u) an arbitrary function. We show that the behaviour ofH(x, u) at spatial infinity determines the causality of M, concretely: (a) if −H(x, u) behaves subquadratically (i.e, essentially −H(x, u) ≤ R0(u)|x| 2−ǫ for some ǫ > 0 and large distance |x| to a fixed point) then the spacetime M is strongly causal, (b) it is globally hyperbolic if, additionally, the spatial part (M0, 〈·, ·〉x) is complete, and (c) M is always causal, but there are non-distinguishing examples (and thus, non-strongly causal), even when −H(x,u) ≤ R0(u)|x| , for any fixed ǫ > 0. Therefore, the classical exact model M0 = R , H(x, u) = ∑ i,j hij(u)xixj( 6≡ 0), which is known to be strongly causal but not globally hyperbolic, lies in the critical quadratic situation with complete M0. This must be taken into account for realistic applications. The relation of these results with the notion of astigmatic conjugacy and the existence of conjugate points is also discussed.