Let K be the space of properly embedded minimal tori in quotients of R^3 by two independant translations, with any fixed (even) number of ends. After an appropriate normalization, we prove that K is a 3-dimensional real analytic manifold that reduces to the finite coverings of the examples defined by Karcher, Meeks and Rosenberg. The degenerate limits of surfaces in K are the catenoid, the helicoid, and three 1-parameter families of surfaces : the simply and doubly Scherk minimal surfaces, and the Riemann minimal examples.