Uniqueness and nonexistence results for complete constant mean curvature spacelike surfaces in certain 3-dimensional Generalized Robertson-Walker spacetimes are shown under assumptions on the extrinsic geometry of the surfaces. In the non-parametric case, all the entire solutions to the constant mean curvature spacelike differential equation are obtained and non-existence results are also shown. The different approaches and techniques used are explained and their ranges of application are discussed. Moreover, comparisons with well-known results in the case of spacelike surfaces in 3-dimensional Lorentz-Minkowski spacetime are exposed.