Non-malleable cryptography, proposed by Dolev, Dwork, and Naor (SICOMP ’00), has numerous applications in protocol composition. In the context of proofs, it guarantees that an adversary who receives a proof cannot maul it into another valid proof. However, non-malleable cryptography (particularly in the non-interactive setting) suffers from an important limitation: An attacker can always copy the proof and resubmit it to another verifier (or even multiple verifiers). In this work, we prevent even the possibility of copying the proof as it is, by relying on quantum information. We call the resulting primitive unclonable proofs, making progress on a question posed by Aaronson. We also consider the related notion of unclonable commitments. We introduce formal definitions of these primitives that model security in various settings of interest. We also provide a near tight characterization of the conditions under which these primitives are possible, including a rough equivalence between unclonable proofs and public-key quantum money.