The directed polymer in a random environment (DPRE) is a model that describes the interaction between a simple random walk and ambient disorder. This model exhibits complex phenomena and transitions from a central limit theory to novel statistical behaviors. Despite extensive study, many aspects and phases remain unidentified. This review focuses on the current understanding of the transition between weak and strong disorder phases, highlights methods developed in the study of the model, and identifies open questions. The DPRE model is defined on $ \mathbb{Z}^d $, with a random walk $ S $ and a family of random variables $ \omega $ representing disorder. The directed polymer measure is a probability measure on random walk paths, weighted by the disorder encountered along the path. The model has physical origins in the study of disordered Ising magnets and has connections to the Kardar-Parisi-Zhang (KPZ) universality class. The model exhibits a transition between weak and strong disorder phases. In the weak disorder regime, the random walk maintains its diffusive behavior, while in the strong disorder regime, localization and superdiffusivity are expected. The critical value $ \beta_c $ separates these phases, and identifying $ \beta_c $ remains a challenging problem. The study of the DPRE model has led to the development of various methods, including martingale methods, combinatorial and representation theoretic methods, and integrable probability techniques. These methods have provided insights into the model's behavior, including the identification of exponents and the derivation of limit theorems. The model has also been studied in the context of heavy-tailed disorder and in higher dimensions. The transition between weak and strong disorder has been analyzed in different dimensions, and the phase diagram for heavy-tailed disorder has been explored. The review highlights the current understanding of the DPRE model, the methods used to study it, and the open questions that remain. It emphasizes the importance of understanding the transition between weak and strong disorder and the role of disorder in shaping the statistical behavior of the model. The review also discusses the connection between the DPRE model and other models in the KPZ universality class, as well as the implications of the model for the broader field of probability theory.The directed polymer in a random environment (DPRE) is a model that describes the interaction between a simple random walk and ambient disorder. This model exhibits complex phenomena and transitions from a central limit theory to novel statistical behaviors. Despite extensive study, many aspects and phases remain unidentified. This review focuses on the current understanding of the transition between weak and strong disorder phases, highlights methods developed in the study of the model, and identifies open questions. The DPRE model is defined on $ \mathbb{Z}^d $, with a random walk $ S $ and a family of random variables $ \omega $ representing disorder. The directed polymer measure is a probability measure on random walk paths, weighted by the disorder encountered along the path. The model has physical origins in the study of disordered Ising magnets and has connections to the Kardar-Parisi-Zhang (KPZ) universality class. The model exhibits a transition between weak and strong disorder phases. In the weak disorder regime, the random walk maintains its diffusive behavior, while in the strong disorder regime, localization and superdiffusivity are expected. The critical value $ \beta_c $ separates these phases, and identifying $ \beta_c $ remains a challenging problem. The study of the DPRE model has led to the development of various methods, including martingale methods, combinatorial and representation theoretic methods, and integrable probability techniques. These methods have provided insights into the model's behavior, including the identification of exponents and the derivation of limit theorems. The model has also been studied in the context of heavy-tailed disorder and in higher dimensions. The transition between weak and strong disorder has been analyzed in different dimensions, and the phase diagram for heavy-tailed disorder has been explored. The review highlights the current understanding of the DPRE model, the methods used to study it, and the open questions that remain. It emphasizes the importance of understanding the transition between weak and strong disorder and the role of disorder in shaping the statistical behavior of the model. The review also discusses the connection between the DPRE model and other models in the KPZ universality class, as well as the implications of the model for the broader field of probability theory.