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It has been observed that transient failures are fairly common in IP backbone networks and there have been several proposals based on local rerouting to provide high network availability despite failures. While most of these proposals are effective in handling single failures, they either cause loops or drop packets in the case of multiple independent failures. To ensure forwarding continuity even with multiple failures, we propose Localized On-demand Link State (LOLS) routing. Under LOLS, each packet carries a blacklist, which is a minimal set of failed links encountered along its path, and the next hop is determined by excluding the blacklisted links. We show that the blacklist can be reset when the packet makes forward progress towards the destination and hence can be encoded in a few bits. Furthermore, blacklist-based forwarding entries at a router can be precomputed for a given set of failures requiring protection. While the LOLS approach is generic, this paper describes how it can be applied to ensure forwarding to all reachable destinations in case of any two link or node failures. Our evaluation of this failure scenario based on various real network topologies reveals that LOLS needs 6 bits in the worst case to convey the blacklist information. We argue that this overhead is acceptable considering that LOLS routing deviates from the optimal path by a small stretch only while routing around failures.