We define a new type of mix network that offers a reduced form of robustness: the mixnet can prove that every message it outputs corresponds to an input submitted by a player without revealing which input (for honest players). We call mixnets with this property reputable mixnets. Reputable mixnets are not fully robust, because they offer no guarantee that distinct outputs correspond to distinct inputs. In particular, a reputable mix may duplicate or erase messages. A reputable mixnet, however, can defend itself against charges of having authored the output messages it produces. This ability is very useful in practice, as it shields the mixnet from liability in the event that an output message is objectionable or illegal. We propose three very efficient protocols for reputable mixnets, all synchronous. The first protocol is based on blind signatures. It works both with Chaumian decryption mixnets or re-encryption mixnets based on ElGamal, but guarantees a slightly weaker form of rep...