AbstractA path P in an edge‐colored graph (not necessarily a proper edge‐coloring) is a rainbow path if no two edges of P are colored the same. For an ℓ‐connected graph G and an integer k with 1 ≤ k ≤ ℓ, the rainbow k‐connectivity rck(G) of G is the minimum integer j for which there exists a j‐edge‐coloring of G such that every two distinct vertices of G are connected by k internally disjoint rainbow paths. The rainbow k‐connectivity of the complete graph Kn is studied for various pairs k, n of integers. It is shown that for every integer k ≥ 2, there exists an integer f(k) such that rck(Kn) = 2 for every integer n ≥ f(k). We also investigate the rainbow k‐connectivity of r‐regular complete bipartite graphs for some pairs k,r of integers with 2 ≤ k ≤ r. It is shown that for each integer k ≥ 2, there exists an integer r such that rck(Kr,r) = 3. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009