  In this paper, we study arithmetically Cohen--Macaulay (ACM) bundles on
homogeneous varieties $G/P$. Indeed we characterize the homogeneous ACM bundles
on $G/P$ of Picard rank one in terms of highest weights. This is a
generalization of the result on $G/P$ of classical types presented by Costa and
Mir\'{o}-Roig for type $A$, and Du, Fang, and Ren for types $B,C$ and $D$. As a
consequence we prove that only finitely many irreducible homogeneous ACM
bundles, up to twisting line bundles, exist over all such $G/P$. Moreover, we
derive the list of the highest weights of the irreducible homogeneous ACM
bundles on particular homogeneous varieties of exceptional types such as the
Cayley Plane and the Freudenthal variety.
