This article presents a comprehensive review of the state-of-the-art advancements in applying artificial intelligence (AI) techniques, especially machine learning and deep learning models, to forecast surface quality in computer numerical control (CNC) machining. Surface quality is influenced by a wide range of factors, which makes its prediction a complex and significant challenge. The factors affecting surface quality are reviewed and categorized into two key elements—tool center positioning errors and the interaction between the tool edge and workpiece materials. As highlighted in recent research of less than five years, the factors are systematically organized into the key elements and presented in tabulated form. Then, particular emphasis is placed on how recent AI techniques have incorporated these factors, addressing the capability of machine learning and deep learning methods to handle the complexity and variability inherent in machining surface quality prediction (MSQP). Moreover, further review is conducted to highlight how advanced AI techniques, particularly transfer learning techniques, have enabled accurate and adaptive MSQP despite data scarcity conditions due to costly experiments and diverse machining conditions. By comprehensively reviewing recent studies from the perspective of the analysis results of key elements affecting surface quality and the inherent characteristics of data-driven AI techniques, this paper identifies the strengths and limitations of various machine learning and deep learning approaches applied in MSQP. Based on the insights into the state of the art, future research directions are discussed for improving prediction accuracy, computational efficiency, and real-time monitoring in the domain.