The cryptanalysis of the cryptographic hash function Tiger has, until now, focussed on finding collisions. In this paper we describe a preimage attack on the compression function of Tiger-12, i.e., Tiger reduced to 12 rounds out of 24, with a complexity of 263.5 compression function evaluations. We show how this can be used to construct second preimages with complexity 263.5 and first preimages with complexity 264.5 for Tiger-12. These attacks can also be extended to Tiger-13 at the expense of an additional factor of 264 in complexity. Key words: Tiger, hash functions, preimages