The interest in triply-periodic minimal surfaces in space seems to date from the work of H. A. Schwarz [11], beginning in 1865 with the construction of the first examples (see w All subsequent work known to us is restricted to these examples. We have found the work of Neovius [14] particularly beautiful and useful. A triply-periodic minimal surface properly immersed in space corresponds to a minimal immersion f of a compact oriented surface X into a fiat 3-torus T. With the induced conformal structure X is a compact Riemann surface and [ is a conformal minimal immersion. Our object is then to study conformal minimal immersions of compact Riemann surfaces in fiat 3-tori. This is also the point of view of Nagano-Smyth [6, 8] and Meeks [5]. The correct setting for this is the Jacobi variety of X and universality (see w plays an indispensable role. The main question studied here is: For a given compact Riemann surface X admitting some conformal minimal immersion f into a flat 3-torus T describe the set of all such immersions. In this set there may be further immersions which are closely related to f and we call them associates of f (see w To describe these we first take a lift of f to universal covers, that is, f:J~---~ R 3. The classical Bonnet deformation gives a one-parameter family of (isometric) conformal minimal immersions f0 :J~---~ R3 (0_<-0 < ~'). For certain values of O these may be triply-periodic and project to conformal minimal immersions of X in flat 3-tori. These projections are the non-trivial associates of f. The great advantage of the Jacobi variety is that these associates are discernible directly from [ and the structure of this variety. This can be found in w The set of conformal minimal immersions of X in flat 3-tori divides into two sets; those having non-trivial associates and those which do not. Theorem 2