Recent years have witnessed significant advancements in cross-chain technology. However, the field faces two pressing challenges. On the one hand, hacks on cross-chain bridges have led to monetary losses of around 3.1 billion USD, highlighting flaws in security models governing interoperability mechanisms and the ineffectiveness of incident response frameworks. On the other hand, users and bridge operators experience restricted privacy, which broadens the potential attack surface.In this paper, we present the most comprehensive study to date on the security and privacy of blockchain interoperability. We employ a systematic literature review, yielding a corpus of 212 relevant documents, including 58 academic papers and 154 gray literature documents, out of a pool of 531 results. We systematically categorize 57 interoperability solutions based on a novel security and privacy taxonomy. Our dataset, comprising academic research, disclosures from bug bounty programs, and audit reports, exposes 45 cross-chain vulnerabilities, 25 theoretical attacks, and 93 mitigation strategies. Leveraging this data, we analyze 14 notable bridge hacks accounting for over 2.9 billion USD in losses, mapping them to the identified vulnerabilities.Our findings reveal that a substantial portion (65.8%) of stolen funds originates from projects secured by intermediary permissioned networks with unsecured cryptographic key operations. Privacy-wise, we demonstrate that achieving unlinkability in cross-chain transactions is contingent on the underlying ledgers providing some form of confidentiality. Our study offers critical insights into the security and privacy of cross-chain systems. We pinpoint promising future research directions, underscoring the urgency of enhancing security and privacy efforts in cross-chain technology. The identified improvements can mitigate the financial risks associated with bridge hacks, fostering user trust in the blockchain ecosystem and, consequently, wider adoption.