Membrane electropermeabilization is the observation that the permeability of a cell membrane can be transiently increased when a micro-millisecond external electric field pulse is applied on a cell suspension or on a tissue. Applicative aspects for the transfer of foreign molecules (macromolecules) into the cytoplasm are routinely used. But only a limited knowledge about what is really occurring in the cell and its membranes at the molecular levels is available. This chapter is a critical attempt to report the present state of the art and to point out some of the still open problems. The experimental facts associated to membrane electropermeabilization are firstly reported. They are valid on biological and model systems. Secondly, soft matter approaches give access to the bioelectrochemical description of the thermodynamical constraints supporting the destabilization of simplified models of the biological membrane. It is indeed described as a thin dielectric leaflet, where a molecular transport takes place by electrophoresis and then diffusion. This naïve approach is due to the lack of details on the structural aspects affecting the living systems as shown in a third part. Membranes are part of the cell machinery. The critical property of cells as being an open system from the thermodynamical point of view is almost never present. Computer simulations are now contributing to our knowledge on electropermeabilization. The last part of this chapter is a (very) critical report of all the efforts that have been performed. The final conclusion remains that we still do not know all the details on the reversible structural and dynamical alterations of the cell membrane (and cytoplasm) supporting its electropermeabilization. We have a long way in basic and translational researches to reach a pertinent description.