  We investigate Banach space automorphisms
$T:\ell_\infty/c_0\rightarrow\ell_\infty/c_0 $ focusing on the possibility of
representing their fragments of the form
$$T_{B,A}:\ell_\infty(A)/c_0(A)\rightarrow \ell_\infty(B)/c_0(B)$$ for $A,
B\subseteq N$ infinite by means of linear operators from $\ell_\infty(A)$ into
$\ell_\infty(B)$, infinite $A\times B$-matrices, continuous maps from
$B^*=\beta B\setminus B$ into $A^*$, or bijections from $B$ to $A$. This leads
to the analysis of general linear operators on $\ell_\infty/c_0$. We present
many examples, introduce and investigate several classes of operators, for some
of them we obtain satisfactory representations and for other give examples
showing that it is impossible. In particular, we show that there are
automorphisms of $\ell_\infty/c_0$ which cannot be lifted to operators on
$\ell_\infty$ and assuming OCA+MA we show that every automorphism of
$\ell_\infty/c_0$ with no fountains or with no funnels is locally, i.e., for
some infinite $A, B\subseteq N$ as above, induced by a bijection from $B$ to
$A$. This additional set-theoretic assumption is necessary as we show that the
continuum hypothesis implies the existence of counterexamples of diverse
flavours. However, many basic problems, some of which are listed in the last
section, remain open.
