The Galois hull of linear code is defined to be the intersection of the code and its Galois dual. In this paper, we investigate the Galois hulls of cyclic codes over F_q^r. For any integer s≤r, we present some sufficient and necessary conditions that cyclic codes have l-dimensional s-Galois hull. Moreover, we prove that a cyclic code C has l-dimensional s-Galois hull iff C has l-dimensional (r-s)-Galois hull. In particular, we also present the sufficient and necessary condition for cyclic codes with 1-dimensional Galois hulls and the relationship between cyclic codes with 1-dimensional Galois hulls and cyclic codes with Galois complementary duals. Some optimal cyclic codes with Galois hulls are obtained. Finally, we explicitly construct a class of cyclic codes with 1-Galois linear complementary dual over F_q³.