Our main contribution is a simple 4 log_2 n-approximation algorithm for the bottleneck asymmetric traveling salesman problem. In addition, we give a constant factor approximation for the restricted version of a problem where every edge lies on a cycle of a bounded length. Finally, we give a new linear programming-based algorithm that achieves a constant factor approximation for the undirected version of the problem. Although the latter is not optimal in terms of the approximation factor, we study it in the hope to gain some insights for the directed graphs.