  The aim of this paper is to classify indecomposable rank 2 arithmetically
Cohen-Macaulay (ACM) bundles on compete intersection Calabi-Yau (CICY)
threefolds and prove the existence of some of them. New geometric properties of
the curves corresponding to rank 2 ACM bundles (by Serre correspondence) are
obtained. These follow from minimal free resolutions of curves in suitably
chosen fourfolds (containing Calabi-Yau threefolds as hypersurfaces). Also the
existence of an indecomposable vector bundle of higher rank on a CICY threefold
of type (2,4) is proved.
