  We compute the renormalization functions on the lattice, in the RI' scheme,
of local bilinear quark operators $\bar{\psi}\Gamma\psi$, where $\Gamma= 1,
\gamma_5, \gamma_\mu, \gamma_5\gamma_\mu, \gamma_5\sigma_{\mu\nu}$. This
calculation is carried out to two loops for the first time. We consider both
the flavor non-singlet and singlet operators.
  As a prerequisite for the above, we compute the quark field renormalization,
$Z_\psi$, up to two loops. We also compute the 1-loop renormalization functions
for the gluon field, $Z_A$, ghost field, $Z_c$, gauge parameter, $Z_\alpha$,
and coupling constant $Z_g$.
  We use the clover action for fermions and the Wilson action for gluons. Our
results are given as an explicit function of the coupling constant, the clover
coefficient $c_{SW}$, and the number of fermion colors ($N_c$) and flavors
($N_f$), in the renormalized Feynman gauge. All 1-loop quantities are evaluated
in an arbitrary gauge.
  Finally, we present our results in the MS-bar scheme, for easier comparison
with calculations in the continuum. We have generalized to fermionic fields in
an arbitrary representation. Some special features of superficially divergent
integrals, obtained from the evaluation of two-loop Feynman diagrams, are
presented in detail in Ref. 1.
