This thesis is concerned with the sequential detection of gradual changes in the location of a stochastic process in two different settings: In Chapter 1 we consider a general stochastic process with a linear drift term which exhibits a possible gradual (non-linear) perturbation at some unknown time point. In Chapter 2 we approach the question of how to detect a gradual change in the location of an unobservable (renewal) process based on observations of the corresponding counting process. We suggest to base the inference on the inverse of the counting process, which behaves similarly as the underlying process itself. In both settings, we introduce detectors and stopping times which follow a common approach on detecting gradual changes: Essentially the idea is to introduce a weight function in order to put less weight on early observations - where a possible change has either not occurred (yet) or is still quite small - and heavy weight on late observations - where a possible change is at its current maximum. This idea is further supported by a (quasi) maximum likelihood approach which suggests to use the assumed type of gradual change as a weighting. Via asymptotic results under the null hypothesis we obtain critical values for the suggested procedures. Under the alternative we show the consistency of the procedures as well as the asymptotic normality of the (standardized) delay times, i.e. the time lag between a change point and its detection.