Semantic communications (SCs) aim to transmit only the essential information required to perform given tasks, thereby improving communication efficiency. Deep learning-based joint source-channel coding (deep JSCC) has emerged as a promising approach for SC systems; however, its performance often degrades when the deployment channels differ from the training channel conditions, making robustness a critical requirement. This paper presents a structured overview of recent methodologies for enhancing the robustness of deep JSCC. Specifically, existing approaches are categorized into two classes: robust training approaches and adaptive approaches, with the latter further divided into adaptive semantic feature selection, physical-layer adaptation, and semantic feature adaptation. Finally, we discuss promising directions, including multi-task generalization and explainability in robust SC systems.