The relevance of a nest of toroidal flux surfaces as a paradigm of the magnetic topology of a tokamak plasma is challenged. High resolution Thomson scattering measurements of electron temperature and density in RTP show several hot filaments in the plasma center and sharp gradients near the sawtooth inversion radius and structures outside the sawtooth region under central ECH. In ohmic plasmas, too, the pressure and temperature profiles show significant bumps. These measurements give evidence of a complex magnetic topology. Transport in a medium with spatially strongly varying diffusivity is considered. It is shown that macroscopic transport is determined by the microscopic structure: a transport theory must predict this structure and the diffusivity in the insulating regions, while the ‘turbulent’ diffusivity is irrelevant. A numerical approach to equilibria with broken surfaces is presented.