Optimal ship routing is crucial for enhancing safety, reducing travel time, and minimizing fuel consumption. This paper introduces and examines recent advancements in stochastic optimization techniques, emerging methods and models for weather-aware ship routing. As marine transportation faces increasing challenges due to climate change and extreme weather events, the need for robust and efficient routing strategies has become imperative. A ship route that is subject to uncertainties is considered stochastic. Therefore, a comprehensive overview of emerging stochastic optimization methods that address the inherent uncertainties in weather forecasting and their impact on optimal routing is presented. The paper explores various approaches, including Markov decision processes, stochastic dynamic programming, and scenario-based optimization, highlights their applications in fuel consumption minimization, ensuring safety and improving time reliability. The integration of ensemble weather forecasts and probabilistic models to capture the stochastic nature of oceanic and atmospheric conditions is discussed. Additionally, computational challenges associated with these methods are analyzed along with recent algorithmic improvements that enhance their scalability and real-time applicability. The inclusion of multiple objectives, such as environmental impact and economic factors, within the stochastic framework is also addressed. Finally, a promising research direction is identified and potential synergies with machine learning techniques to further account for an increasingly uncertain marine environment.