Recent research has highlighted the importance of domain-general processes and brain regions for language and semantic cognition. Yet, this has been mainly observed in executively demanding tasks, leaving open the question of the contribution of domain-general processes to natural language and semantic cognition. Using fMRI, we investigated whether neural processes reflecting context integration and context update-two key aspects of naturalistic language and semantic processing-are domain-specific versus domain-general. Thus, we compared neural responses during the integration of contextual information across semantic and non-semantic tasks. Whole-brain results revealed both shared (left posterior-dorsal inferior frontal gyrus, left posterior inferior temporal gyrus, and left dorsal angular gyrus/intraparietal sulcus) and distinct (left anterior-ventral inferior frontal gyrus, left anterior ventral angular gyrus, left posterior middle temporal gyrus for semantic control only) regions involved in context integration and update. Furthermore, data-driven functional connectivity analysis clustered domain-specific versus domain-general brain regions into distinct but interacting functional neural networks. These results provide a first characterisation of the neural processes required for context-dependent integration during language processing along the domain-specificity dimension, and at the same time, they bring new insights into the role of left posterior lateral temporal cortex and left angular gyrus for semantic cognition.