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In this paper, our goal is to adapt a pre-trained convolutional neural network to domain shifts at test time. We do so continually with the incoming stream of test batches, without labels. The existing literature mostly operates on artificial shifts obtained via adversarial perturbations of a test image. Motivated by this, we evaluate the state of the art on two realistic and challenging sources of domain shifts, namely contextual and semantic shifts. Contextual shifts correspond to the environment types, for example, a model pre-trained on indoor context has to adapt to the outdoor context on CORe-50. Semantic shifts correspond to the capture types, for example a model pre-trained on natural images has to adapt to cliparts, sketches, and paintings on DomainNet. We include in our analysis recent techniques such as Prediction-Time Batch Normalization (BN), Test Entropy Minimization (TENT) and Continual Test-Time Adaptation (CoTTA). Our findings are three-fold: i) Test-time adaptation methods perform better and forget less on contextual shifts compared to semantic shifts, ii) TENT outperforms other methods on short-term adaptation, whereas CoTTA outpeforms other methods on long-term adaptation, iii) BN is most reliable and robust. Our code is available at https://github.com/tommiekerssies/Evaluating-Continual-Test-Time-Adaptation-for-Contextual-and-Semantic-Domain-Shifts.