In a process for monitoring a computer network with at least two subscribers connected via a data bus comprising at least two lines, each subscriber having a receiving and/or a sending part, the function of the data bus and/or the function of the subscribers monitors at least one subscriber using watch-dog signals (CEI),a nd the emergency running measures (M1; M2) appropriate to the error in question are adopted in order to set up defined emergency operating modes (single-wire, special emergency running). After adoption of the first emergency measure (M1), the computer network is tested for errors. If no errors are detected, the first emergency running measure is maintained and hence the resultant emergency operating mode (single-wire emergency). If other errors occur in the computer network after adoption of the first emergency running measure, the first emergency measure is withdrawn and the second emergency running measure (M2) is adopted, i.e., special emergency running, is maintained. Only if other errors then occur are the subscribers concerned or the whole network switched off (bus-off).