  Rank 2 indecomposable arithmetically Cohen-Macaulay bundles E on a
nonsingular cubic surface X in P^3 are classified, by means of the possible
forms taken by the minimal graded free resolution of E over P^3.
  The admissible values of the Chern classes of E are listed and the vanishing
locus of a general section of E is studied.
  Properties of E such as (semi) stability and simplicity are investigated; the
number of relevant families is computed together with their dimension.
