  The effects of the final state interaction (FSI) in semi inclusive deep
inelastic electron scattering processes $A(e,e'p)X$ off nuclei are investigated
in details. Proton production is described within the spectator and the target
fragmentation mechanisms whose relevance to the experimental study of the deep
inelastic structure functions of bound nucleons and the non perturbative
hadronization process is analyzed. Particular attention is paid to the deuteron
target within kinematical conditions corresponding to the available and
forthcoming experimental data at Jlab.
  We argue that there are kinematical regions where FSI effects are minimized,
allowing for a reliable investigation of the DIS structure functions, and
regions where the interaction of the quark-gluon debris with nucleons is
maximized, which makes it possible to study hadronization mechanisms. Nuclear
structure has been described by means of realistic wave functions and spectral
functions and the final state interaction has been treated within an eikonal
approximation approach which takes into account the rescattering of the
quark-gluon debris with the residual nucleus and, in the case of complex
nuclei, within an optical potential approach to account for the FSI of the
struck proton.
