Deep learning (DL) has emerged as a transformative technology with immense potential to reshape the sixth-generation (6G) wireless communication network. By utilizing advanced algorithms for feature extraction and pattern recognition, DL provides unprecedented capabilities in optimizing the network efficiency and performance, particularly in physical layer communications. Although DL technologies present the great potential, they also face significant challenges related to the robustness, which are expected to intensify in the complex and demanding 6G environment. Specifically, current DL models typically exhibit substantial performance degradation in dynamic environments with time-varying channels, interference of noise and different scenarios, which affect their effectiveness in diverse real-world applications. This paper provides a comprehensive overview of strategies and approaches for robust DL-based methods in physical layer communications. First we introduce the key challenges that current DL models face. Then we delve into a detailed examination of DL approaches specifically tailored to enhance robustness in 6G, which are classified into data-driven and model-driven strategies. Finally, we verify the effectiveness of these methods by case studies and outline future research directions.