  In this paper we derive a list of all the possible indecomposable normalized
rank--two vector bundles without intermediate cohomology on the prime Fano
threefolds and on the complete intersection Calabi Yau threefolds, say $V$, of
Picard number $\rho=1$. For any such bundle $\E$, if it exists, we find the
projective invariants of the curves $C \subset V$ which are the zero-locus of
general global sections of $\E$. In turn, a curve $C \subset V$ with such
invariants is a section of a bundle $\E$ from our lists. This way we reduce the
problem for existence of such bundles on $V$ to the problem for existence of
curves with prescribed properties contained in $V$. In part of the cases in our
lists the existence of such curves on the general $V$ is known, and we state
the question about the existence on the general $V$ of any type of curves from
the lists.
