  A bounded linear operator between Banach spaces is called {\it completely
continuous} if it carries weakly convergent sequences into norm convergent
sequences. Isolated is a universal operator for the class of
non-completely-continuous operators from $L_1$ into an arbitrary Banach space,
namely, the operator from $L_1$ into $\ell_\infty$ defined by $$ T_0 (f)
=\left( \int r_n f \, d\mu \right)_{n\ge 0} \ , $$ where $r_n$ is the
$n^{\text{th}}$ Rademacher function. It is also shown that there does not exist
a universal operator for the class of non-completely-continuous operators
between two arbitrary Banach space. The proof uses the factorization theorem
for weakly compact operators and a Tsirelson-like space.
