In the literature there is a variety of models for the Calvin cycle of photosynthesis and a number of statements, based on computer calculations, about the number and stability of their steady states. Here I will discuss some progress in obtaining rigorous results on this issue. It has been shown that in a simple model with five unknowns there are parameters for which there are two positive steady states, one of them stable, and a special case where there is a continuum of steady states. A more detailed model, due to Pettersson and Ryde-Pettersson, with fifteen unknowns turns out to be a system of differential-algebraic equations (DAE), rather than a system of ODE. It has been shown that for certain parameters it has two positive steady states. The proof is related to the method of elementary flux modes. For a closely related model, due to Poolman, there are three positive steady states. The difference between the numbers of steady states in these two models is related to the biological phenomenon of overload breakdown, in which too much demand for sugar causes the system supplying it to collapse.