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I've moved into a new flat (apartment).
It has these small scrolls in containers on door frames outside and inside the apartment.
I am not Jewish.
I do want to be respectful of other people's beliefs.
What are these scrolls called? What should I do with them? These particular versions seem to be really nicely crafted.
It is called a Mezuza. Contact the local Chabad House Rabbi and he will come pick them up and dispose of them appropiately.
Depending on their composition and state, they might possibly be worth some money. If they're on paper, they're worthless. If they're on parchment, though, then it depends on whether they're still "valid", which the local rabbi should be able to discern. If they are still valid, then they may might have some value depending on the condition. If so, you'd be within your rights to ask the rabbi to buy them from you at a reasonable "used" price. (thanks to msh210 and Isaac Moses for this edit)
I agree with the recommendation in this answer to consult a local rabbi for specific guidance.
But before you dispose of them: You said that you're in an apartment and that the cases are really nice. It is possible that these were not abandoned by the previous tenant but are instead the property of the landlord, who might believe he's required (as a Jew) to place them in his property and not just his own dwelling. So before you sell these to the local rabbi only to find out when your lease ends that your landlord expected them to still be there, I would strongly advise consulting the landlord. He might choose to remove them himself. Or he might be a gentile who has no idea the previous tenant left them and he doesn't care.
