This paper synthesizes the latest advancements in solving the Optimal Power Flow (OPF) problem within power systems through the application of machine learning, specifically neural network technologies. The OPF problem, crucial for efficient and reliable power system operation, involves minimizing generation costs while adhering to a variety of technical and physical constraints. This study highlights innovative learning-based OPF approaches, including End-to-End (E2E) and Learning-to-Optimize (L2O) methods. These strategies are designed to lessen the computational load of online optimization by leveraging extensive offline training with historical data to either predict outcomes or enhance the performance of traditional optimizers. The effectiveness of these methods in real-world settings is evaluated, and potential avenues for future research are discussed.