The model of directed polymers in a random environment (DPRE) is a fundamental model that explores the interaction between a simple random walk and ambient disorder. This interaction leads to complex phenomena, transitioning from 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 motivated by the model, and identifies open questions. The DPRE is defined on \(\mathbb{Z}^d\) with a random walk \(S = (S_n)_{n \geq 0}\) and a family of random variables \(\omega = (\omega_{n,x})_{n \in \mathbb{N}, x \in \mathbb{Z}^d}\). The directed polymer measure \(\mu_{n,x}^{\beta,\omega}\) is a probability measure on the random walk paths, weighted by the cumulative disorder up to time \(n\). The partition function \(Z_n^{\beta,\omega}(x)\) normalizes this measure. The transition between weak and strong disorder phases is marked by a critical value \(\beta_c\), where weak disorder occurs if \(W_\infty^\beta > 0\) almost surely, and strong disorder occurs if \(W_\infty^\beta = 0\) almost surely. The weak disorder regime is characterized by uniform integrability of the partition function \(W_n^\beta\), which implies that \(\mathbb{E}[W_\infty^\beta] = 1\). Key methods in studying the DPRE include martingale approaches, fractional moment methods, and coarse graining. The weak disorder regime has been well-studied, with results on central limit theorems and the weak disorder regime extending to higher dimensions. However, the strong disorder regime and the critical dimension 2 remain challenging, with recent progress in understanding hidden phase transitions and the intermediate disorder regime. The review also discusses the connection to random walk pinning models and the non-coincidence of \(\beta_2\) and \(\beta_c\) in higher dimensions. The critical value \(\beta_c\) is determined through moment estimates and the fractional moment method, which has been crucial in establishing strong disorder and very strong disorder conditions. Open questions include the equivalence of strong and very strong disorder for unbounded disorder, the behavior of the derivative of the free energy in the strong disorder regime, and the detailed localization properties in the log-gamma polymer model.The model of directed polymers in a random environment (DPRE) is a fundamental model that explores the interaction between a simple random walk and ambient disorder. This interaction leads to complex phenomena, transitioning from 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 motivated by the model, and identifies open questions. The DPRE is defined on \(\mathbb{Z}^d\) with a random walk \(S = (S_n)_{n \geq 0}\) and a family of random variables \(\omega = (\omega_{n,x})_{n \in \mathbb{N}, x \in \mathbb{Z}^d}\). The directed polymer measure \(\mu_{n,x}^{\beta,\omega}\) is a probability measure on the random walk paths, weighted by the cumulative disorder up to time \(n\). The partition function \(Z_n^{\beta,\omega}(x)\) normalizes this measure. The transition between weak and strong disorder phases is marked by a critical value \(\beta_c\), where weak disorder occurs if \(W_\infty^\beta > 0\) almost surely, and strong disorder occurs if \(W_\infty^\beta = 0\) almost surely. The weak disorder regime is characterized by uniform integrability of the partition function \(W_n^\beta\), which implies that \(\mathbb{E}[W_\infty^\beta] = 1\). Key methods in studying the DPRE include martingale approaches, fractional moment methods, and coarse graining. The weak disorder regime has been well-studied, with results on central limit theorems and the weak disorder regime extending to higher dimensions. However, the strong disorder regime and the critical dimension 2 remain challenging, with recent progress in understanding hidden phase transitions and the intermediate disorder regime. The review also discusses the connection to random walk pinning models and the non-coincidence of \(\beta_2\) and \(\beta_c\) in higher dimensions. The critical value \(\beta_c\) is determined through moment estimates and the fractional moment method, which has been crucial in establishing strong disorder and very strong disorder conditions. Open questions include the equivalence of strong and very strong disorder for unbounded disorder, the behavior of the derivative of the free energy in the strong disorder regime, and the detailed localization properties in the log-gamma polymer model.