  We review recent progress on ancestral processes related to
mutation-selection models, both in the deterministic and the stochastic
setting. We mainly rely on two concepts, namely, the killed ancestral selection
graph and the pruned lookdown ancestral selection graph. The killed ancestral
selection graph gives a representation of the type of a random individual from
a stationary population, based upon the individual's potential ancestry back
until the mutations that define the individual's type. The pruned lookdown
ancestral selection graph allows one to trace the ancestry of individuals from
a stationary distribution back into the distant past, thus leading to the
stationary distribution of ancestral types. We illustrate the results by
applying them to a prototype model for the error threshold phenomenon.
