We consider the properties of charged domain walls in ferroelectrics as a quantum problem. This includes determination of self-consistent attracting 1D potential for compensating charge carriers, the number and positions of discrete energy levels in this potential, dependencies on the ferroelectric characteristics, as well as the spatial structure and formation energy of the wall. Our description is based on the Hartree and Thomas-Fermi methods and Landau theory for the ferroelectric transitions. Changeover from a few to many quantum levels (with the electron binding energies similar to 1 eV) is controlled by a single characteristic parameter. The quantum models well describe the core of the wall, whose width is typically similar to 10 nm. Additionally, the walls possess pronounced long-range tails which are due to trap recharging. For the trap concentration N-t = (10(17)-10(18)) cm(-3), the tail length l is of the mu m scale. On the distances much larger than l the walls are electrically uncoupled from each other and the crystal faces.