In recent years several authors have obtained limit theorems for L n , the location of the rightmost particle in a supercritical branching random walk but all of these results have been proved under the assumption that the offspring distribution has ϕ(θ) = ∝ exp(θx)dF(x) 0. In this paper we investigate what happens when there is a slowly varying function K so that 1−F(x)∼x }-q K(x) as x → ∞ and log(−x)F(x)→0 as x→−∞. In this case we find that there is a sequence of constants a n , which grow exponentially, so that L n /a n converges weakly to a nondegenerate distribution. This result is in sharp contrast to the linear growth of L n observed in the case ϕ(θ)<∞.