In this paper we will derive asymptotically optimal rank-order tests for independence against suitable classes of nonparametric alternatives and give asymptotic relative efficiencies (ARE's) of such tests under general contiguous alternatives of positive quadrant dependence (cf. Lehmann [14]). From Lehmann [14] one can also see that such alternatives in some respects are more general than the alternatives considered in Bhuchongkul [1], Konijn [12], Hajek and Sidak [8] page 221), and others. For the problem of symmetry and for the two-sample problem we can get completely analogous results with similar proofs. Details are omitted. The paper is based on the theory of contiguity that was introduced by LeCam [13] and Hajek [6]. The results of this paper complement results obtained by Hodges and Lehmann [9], [10], Chernoff and Savage [3], Hajek [6], van Eeden [4], Bhuchongkul [1], Gokhale [5], and others.