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Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topicin Lorentzian conformal geometry which parallels the theory of Willmoresurfaces in $S^4$, are studied in this paper. We define two kinds of transformsfor such a surface, which produce the so-called left/right polar surfaces andthe adjoint surfaces. These new surfaces are again conformal Willmore surfaces.For them holds interesting duality theorem. As an application spacelikeWillmore 2-spheres are classified. Finally we construct a family of homogeneousspacelike Willmore tori.