We will report on our recent work on complete spacelike minimal surfaces in 4-dimensional Lorentz space.  Our method is still by complex functions and Weierstrass representation. Compared with classical minimal surfaces, the first difference is that there exist complete examples with finite total curvature yet it is non-algebraic (i.e. the Gauss maps can not extend to the ends). The second new phenomenen is certain singular ends where the total curvature might converge (called good singular end, with well-defined integer-valued index) or diverge. This is a joint work with Zhiyu Liu, Changping Wang, and Peng Wang .