Obreht sets up the story with three separate threads: the first, set in the present, features a young doctor, Natalia, grief-stricken at the loss of her beloved grandfather. The second tracks her grandfather's long acquaintance with the "deathless man," a mysterious traveller who Natalia thinks might have led her grandfather on his final journey away from home. The third thread, set during the grandfather's childhood, is that from which the title of the book is drawn (and you'll just have to read it for more, because I'm not going to spoil it for you).
I read this book extremely (and uncharacteristically) slowly, mostly because I didn't want to be done with it. It's clearly a labor of love, and the way Obreht subtly connects her three threads, each so very different from the others, made for excellent reading. I enjoyed her ability to shift between magical tales and the gritty reality of the post-conflict Balkans, as well as the fascinating characters she's created and the captivating stories she tells.
Highly recommended, and keep your eye on Obreht. I feel confident there's more where this came from!