We study the k-route cut problem: given an undirected edge-weighted graph G=(V,E), a collection {(s_1,t_1),(s_2,t_2),...,(s_r,t_r)} of source-sink pairs, and an integer connectivity requirement k, the goal is to find a minimum-weight subset E' of edges to remove, such that the connectivity of every pair (s_i, t_i) falls below k. Specifically, in the edge-connectivity version, EC-kRC, the requirement is that there are at most (k-1) edge-disjoint paths connecting s_i to t_i in G \ E', while in the vertex-connectivity version, NC-kRC, the same requirement is for vertex-disjoint paths. Prior to our work, poly-logarithmic approximation algorithms have been known for the special case where k >= 3, but no non-trivial approximation algorithms were known for any value k>3, except in the single-source setting. We show an O(k log^{3/2}r)-approximation algorithm for EC-kRC with uniform edge weights, and several polylogarithmic bi-criteria approximation algorithms for EC-kRC and NC-kRC, where the connectivity requirement k is violated by a constant factor. We complement these upper bounds by proving that NC-kRC is hard to approximate to within a factor of k^{eps} for some fixed eps>0. We then turn to study a simpler version of NC-kRC, where only one source-sink pair is present. We give a simple bi-criteria approximation algorithm for this case, and show evidence that even this restricted version of the problem may be hard to approximate. For example, we prove that the single source-sink pair version of NC-kRC has no constant-factor approximation, assuming Feige's Random k-AND assumption.