Suppose that a sequence of data points follows a distribution of a certain parametric form, but that one or more of the underlying parameters may change over time. This paper addresses various natural questions in such a framework. We construct canonical monitoring processes which under the hypothesis of no change converge in distribution to independent Brownian bridges, and use these to construct natural goodness-of-fit statistics. Weighted versions of these are also studied, and optimal weight functions are derived to give maximum local power against alternatives of interest. We also discuss how our results can be used to pinpoint where and what type of changes have occurred, in the event that initial screening tests indicate that such exist. Our unified large-sample methodology is quite general and applies to all regular parametric models, including regression, Markov chains, and time series situations.