Among many properties distinguishing emergence, such as novelty, irreducibility and unpredictability, computational accounts of emergence in terms of computational incompressibility aim first at making sense of such unpredictability. Those accounts prove to be more objective than usual accounts in terms of levels of mereology, which often face objections of being too epistemic. The present paper defends computational accounts against some objections, and develops what such notions bring to the usual idea of unpredictability. I distinguish the objective unpredictability, compatible with determinism (...) and entailed by emergence, and various possibilities of predictability at emergent levels. This makes sense of practices common in complex systems studies that forge qualitative predictions on the basis of comparisons of simulations with multiple values of parameters. I consider robustness analysis as a way to ensure the ontological character of computational emergence. Finally, I focus on the property of novelty, as it is displayed by biological evolution, and ask whether computer simulations of evolution can produce the same kind of emergence as the open-ended evolution attested in Phanerozoic records. (shrink)