The maritime industry is integral to global trade and heavily depends on precise forecasting to maintain efficiency, safety, and economic sustainability. Adopting deep learning for predictive analysis has markedly improved operational accuracy, cost efficiency, and decision-making. This technology facilitates advanced time series analysis, vital for optimizing maritime operations. This paper reviews deep learning applications in time series analysis within the maritime industry, focusing on three areas: ship operation-related, port operation-related, and shipping market-related topics. It provides a detailed overview of the existing literature on applications such as ship trajectory prediction, ship fuel consumption prediction, port throughput prediction, and shipping market prediction. The paper comprehensively examines the primary deep learning architectures used for time series forecasting in the maritime industry, categorizing them into four principal types. It systematically analyzes the advantages of deep learning architectures across different application scenarios and explores methodologies for selecting models based on specific requirements. Additionally, it analyzes data sources from the existing literature and suggests future research directions.