Multi-party fair exchange (MFE) and fair secure multi-party computation (fair SMPC) are under-studied fields of research, with practical importance. We examine MFE scenarios where every participant has some item, and at the end of the protocol, either every participant receives every other participant’s item, or no participant receives anything. This is a particularly hard scenario, even though it is directly applicable to protocols such as fair SMPC or multi-party contract signing. We further generalize our protocol to work for any exchange topology. We analyse the case where a trusted third party (TTP) is optimistically available, although we emphasize that the trust put on the TTP is only regarding the fairness, and our protocols preserve the privacy of the exchanged items even against a malicious TTP. We construct an asymptotically optimal (for the complete topology) multi-party fair exchange protocol that requires a constant number of rounds, in comparison to linear, and O(n^2) messages, in comparison to cubic, where n is the number of participating parties. We enable the parties to efficiently exchange any item that can be efficiently put into a verifiable escrow (e.g., signatures on a contract). We show how to apply this protocol on top of any SMPC protocol to achieve a fairness guarantee with very little overhead, especially if the SMPC protocol works with arithmetic circuits. Our protocol guarantees fairness in its strongest sense: even if all n−1 other participants are malicious and colluding, fairness will hold.