We describe a collision-finding attack on 16 rounds of the Tiger hash function requiring the time for about 244 compression function invocations. This extends to a collision-finding attack on 17 rounds of the Tiger hash function in time of about 249 compression function invocations. Another attack generates circular near-collisions, for 20 rounds of Tiger with work less than that of 249 compression function invocations. Since Tiger has only 24 rounds, these attacks may raise some questions about the security of Tiger. In developing these attacks, we adapt the ideas of message modification attacks and neutral bits, developed in the analysis of MD4 family hashes, to a completely different hash function design.