Counter-rotating discs can arise from the accretion of a counter-rotating gas cloud on to the surface of an existing corotating disc or from the counter-rotating gas moving radially inwards to the outer edge of an existing disc. At the interface, the two components mix to produce gas or plasma with zero net angular momentum which tends to free-fall towards the disc centre. We discuss high-resolution axisymmetric hydrodynamic simulations of viscous counter-rotating discs for the cases where the two components are vertically separated and radially separated. The viscosity is described by an isotropic ...-viscosity including all terms in the viscous stress tensor. For the vertically separated components, a shear layer forms between them and the middle part of this layer free-falls to the disc centre. The accretion rates are increased by factors of ~10...-10... over that for a conventional disc rotating in one direction with the same viscosity. The vertical width of the shear layer and the accretion rate are strongly dependent on the viscosity and the mass fraction of the counter-rotating gas. In the case of radially separated components where the inner disc corotates and the outer disc rotates in the opposite direction, a gap between the two components opens and closes quasi-periodically. The accretion rates are ...25 times larger than those for a disc rotating in one direction with the same viscosity. (ProQuest: ... denotes formulae/symbols omitted.)