  This contribution is concerned with mathematical models for the dynamics of
the genetic composition of populations evolving under recombination.
Recombination is the genetic mechanism by which two parent individuals create
the mixed type of their offspring during sexual reproduction. The corresponding
models are large, nonlinear dynamical systems (for the deterministic treatment
that applies in the infinite-population limit), or interacting particle systems
(for the stochastic treatment required for finite populations). We review
recent progress on these difficult problems. In particular, we present a closed
solution of the deterministic continuous-time system, for the important special
case of single crossovers; we extract an underlying linearity; we analyse how
this carries over to the corresponding stochastic setting; and we provide a
solution of the analogous deterministic discrete-time dynamics, in terms of its
generalised eigenvalues and a simple recursion for the corresponding
coefficients.
