We study the behavior of the random walk on the infinite cluster of independent long range percolation in dimensions d=1,2, where x and y are connected with probability \beta/|x-y|^{-s}. We show that when s is between d and 2d, the walk is transient, and when s is greater than or equal to 2d, the walk is recurrent. The proof of transience is based on a renormalization argument. As a corollary of this renormalization argument, we get that for every dimension d, if s is between d and 2d, then critical percolation has no infinite clusters. This result is extended to the free random cluster model. A second corollary is that when d is at least 2 and, again, s is between d and 2d, we can erase all long enough bonds and still have an infinite cluster. The proof of recurrence in two dimensions is based on general stability results for recurrence in random electrical networks. In particular, we show that i.i.d. conductances on a recurrent graph of bounded degree yield a recurrent electrical network.