The authors analyse four interacting random-walk models from a comparative point of view, concentrating on one-dimensional versions. They consider an interacting random-walk model recently introduced by Stanley et al. (1983) the Domb-Joyce model (1972), and the 'true' self-avoiding walk of Amit et al. (1983). In addition, they introduce a model based on the weighting of turning points, whose properties are related to those of the Ising spin chain. Fixed points and their stability are identified for both attractive and repulsive correlations in all four cases. They compare the mechanisms involved in each of these models and comment on the role played in each by its particular form of correlation between steps. The different universal behaviours are shown to arise from competition between short-range and long-range (and particularly cumulative) memory effects, and between local and global normalisation conditions.