Asymptotic methods based on the slenderness ratio are used to obtain the leading-order equations which govern the fluid dynamics of axisymmetric, isothermal, Newtonian, compound liquid jets such as those employed in the manufacture of textile fibres, composite fibres and optical fibres, at low Reynolds numbers. It is shown that the leading-order equations are one-dimensional, and analytical solutions are obtained for steady flows at zero Reynolds numbers, zero gravitational pull, and inertialess jets. A linear stability analysis of the viscous flow regime indicates that the stability of compound jets is governed by the same eigenvalue equation as that for the spinning of round fibres and annular jets. Numerical studies of the time-dependent equations subject to axial velocity perturbations at either the nozzle exit or the take-up point, or both, indicate that the compound jet dynamics evolves from periodic to chaotic motions as the extension or draw ratio is increased. The power spectrum of the inner (round) jet's radius at the take-up point broadens and the phase diagrams exhibit holes at large draw ratios. The number of holes increases as the draw ratio is increased, thus indicating chaotic behaviour. It is also shown that the nonlinear dynamics of bicomponent, compound jets is analogous to that of single-component, annular jets.