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We review recent progress in the understanding of the role of multiple- andsimultaneous multiple merger coalescents as models for the genealogy inidealised and real populations with exceptional reproductive behaviour. Inparticular, we discuss models with `skewed offspring distribution' (or underother non-classical evolutionary forces) which lead in the single locus haploidcase to multiple merger coalescents, and in the multi-locus diploid case tosimultaneous multiple merger coalescents. Further, we discuss inference methodsunder the infinitely-many sites model which allow both model selection andestimation of model parameters under these coalescents.

Genealogies and inference for populations with highly skewed offspring distributions