Using the example of changes in the mean we introduce change point estimators for multiple changes based on moving sum statistics. For a given bandwidth we prove consistency of these estimators as well as their asymptotic limit distribution. In many practical situations a single bandwidth is not sufficient to detect all change points, but instead several bandwidths need to be used. We discuss extensions based on information criteria to combine this multiscale information in a consistent way. Finally, extensions to a very general setting based on estimating functions are discussed. These extensions allow for more robust methods in the mean change situation as well as for more complex change point problems such as changes in the time series structure.