  In this paper, we work on the marginally trapped surfaces in the
4-dimensional Minkowski, de Sitter and anti-de Sitter space-times. We obtain
the complete classification of the marginally trapped surfaces in the Minkowski
space-time with pointwise 1-type Gauss map. Further, we give construction of
marginally trapped surfaces with 1-type Gauss map and a given boundary curve.
We also state some explicit examples. We also prove that a marginally trapped
surface in the de Sitter space-time $\mathbb S^4_1(1)$ or anti-de Sitter
space-time $\mathbb H^4_1(-1)$ has pointwise 1-type Gauss map if and only if
its mean curvature vector is parallel. Moreover, we obtain that there exists no
marginally trapped surface in $\mathbb S^4_1(1)$ or $\mathbb H^4_1(-1)$ with
harmonic Gauss map.
