The spectral curve of a conformal torus Classically, a spectral curve can be associated to tori which are given by a harmonicity condition, such as constant mean curvature tori or Willmore tori. The harmonicity allows to introduce a spectral parameter and one obtains an associated family of flat connections: the spectral curve is then essentially given by the eigenvalues of the holonomies of the flat connections. Recently, a more general notion of a spectral curve has been introduced for any conformally immersed torus in the 4-sphere. We will give a geometric interpretation of the spectral curve in terms of Darboux transforms and will illustrate the construction in the case of a constant mean curvature torus.