I have an Italian resident permit (for humanity reasons).
Can I travel around the EU with this resident permit, without a passport?
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You won't be able to travel legally with only the residence permit (permesso di soggiorno). You also need your titolo di viaggio per stranieri (refugee travel document). This passport-replacing document will allow you to visit other Schengen states visa-free for 90-in-180 days, and also apply for visas to visit any other countries, for example the UK or USA.
This travel document should have been issued to you at the same time you received your residence permit. If you did not receive it, visit your nearest police station to apply for it.