  Dark energy and dark matter are only indirectly measured via their
gravitational effects. It is possible that there is an exchange of energy
within the dark sector, and this offers an interesting alternative approach to
the coincidence problem. We consider two broad classes of interacting models
where the energy exchange is a linear combination of the dark sector densities.
The first class has been previously investigated, but we define new variables
and find a new exact solution, which allows for a more direct, transparent and
comprehensive analysis. The second class has not been investigated in general
form before. We give general conditions on the parameters in both classes to
avoid unphysical behavior (such as negative energy densities).
