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).