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Time series forecasting represents one of the most critical challenges in contemporary data science and machine learning, with applications spanning finance, energy systems, weather prediction, traffic management, supply chain optimization, and healthcare. This comprehensive review examines and compares three prominent forecasting methodologies: Autoregressive Integrated Moving Average (ARIMA), Long Short-Term Memory (LSTM) neural networks, and Prophet. These models embody distinct paradigms—traditional statistical methods, deep learning architectures, and automated trend-based analysis respectively. Through systematic synthesis of recent literature and empirical studies from 2018–2025, this review analyzes theoretical foundations, practical implementations, strengths, limitations, and optimal application contexts. Our findings reveal that ARIMA exhibits superior performance for simple linear patterns (MAPE 3.2–13.6%), LSTM demonstrates exceptional capability in capturing complex non-linear dependencies with 84–87% error reduction vs. ARIMA, while Prophet excels in handling business time series with strong seasonality (MAPE 2.2–24.2%). Model selection depends critically on data characteristics, forecasting horizon, computational resources, and application requirements. This review synthesizes over two decades of empirical findings to provide principled guidance for practitioners in model selection and implementation.