Circular data and their applications are pervasive across numerous disciplines. In the field of circular data analytics and statistical learning, distributional studies, regression and time series models have played crucial roles. In this paper, we present a comprehensive framework for circular regression and time series models. We employ the maximum likelihood estimation approach coupled with local and global influence methods to provide a robust methodology. Furthermore, we explore several specific models for analysing circular data and investigate techniques to identify influential observations within these models. To demonstrate the effectiveness of our proposed methods, we provide simulated and real data examples.