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Death is our guide and narrator to The Book Thief. But don't go cowering under the covers on his account. For one thing, Death ain't the bad guy (that would be Hitler). For another, Death is in some ways pretty human—especially considering how inhumane a lot of the homo sapiens running around in The Book Thief are.
For example, Death has real feelings. We see him experience both sadness and joy in the novel. He even gets depressed. It seems like the poor guy hasn't had a vacation since he started working. He might not even have had a coffee break.
And, let's face it, his job is about as depressing as it gets. To help distract him from his sad and never-ending work, he often fixates on the color of the sky at the time of each human death.
Like many humans, Death tries to find ways to give meaning to his work. One of the main things he does is collect stories of courageous humans. Liesel is particularly interesting to him because of her courage and her personality. Stories like hers help keep him going.
He retells these stories, he says, "to prove to myself that you, and your human existence, are worth it" (4.33). In other words, he looks for hope in the gathering, reading, and telling of stories. This quest for meaning seems like a very human thing indeed.
On the other hand, though Death has lots of human qualities, he'll never be totally human. (Tough break, Death.) In the end, he's still the being that comes to separate our souls from our bodies, Death's final line, "I'm haunted by humans" (88.17), shows that he'll always be separate from us. He tells us at the beginning of the novel that the most painful part of his job is seeing "[t]he survivors," "the leftover humans," "the ones who are left behind, crumbling among the jigsaw puzzle of realization, despair, and surprise" (1.20, 1.21, 1.22). You could argue that Death is in the unlucky position of having human-like emotions, but never being allowed to be a part of humanity.
Of course, though Death may not be an actual human being, he's totally linked to all of humanity. It's important to note that he doesn't cause people to die, but is "a result" (2.3). He exists because people die, to help them transition to the afterlife. Also, note that Death isn't your typical grim reaper with "sickle or scythe" (45.2). He tells us that if we want to see what he looks like, we should, "[f]ind […] a mirror" (45.2).
In other words, all humans die, and so we all look like Death. In a way, we're all united with Death, and he's the thing that unites all of us. He's part of what makes us essentially human. Oof. This just got scary-deep.
Now, check out what Markus Zusak himself has to say about his oh-so-humane version of Death:
I guess there's a little bit of death in me, but it's probably true for everyone. I think I just applied the thought of how scared I am of death and reversed it. I thought, 'What if he or she or it is haunted by everything he sees humans do?' In that way, he's also like all of us, because we all have the same reactions to each other's behavior. Also, I had more empathy for Death when he was vulnerable like that. (Source)
We have another question: where does Death take the souls? This isn't quite specified... though he does seem to cross the sky to get to wherever he's going. A conception of the afterlife isn't fleshed out, though Death seems to know the secret and drops some hints. He's elusive about the afterlife, but does seem to imply that there is one.
When he comes for Liesel years into the future, he takes her to Anzac Avenue—which sounds ordinary enough, but maybe Death means an afterlife version? It's confusing and playful, and made even more confusing when death tells us that, "A few cars drove by, each way. Their drivers were Hitlers and Hubermanns, and Maxes, killers, Dillers, and Steiners…" (88.13). Those are all names of the dead. What's surprising is that everybody, including Hitler, seems to be in the same place.
This means that Death probably doesn't believe that what we do in this life necessarily impacts where we go when we die. As far as Death is concerned, living a good or bad life is its own reward or punishment. If you have strong beliefs regarding the afterlife, this idea of Death as a great equalizer might really strike a nerve.