This paper presents a comprehensive review of the historical development, the current state of the art, and prospects of data-driven approaches for industrial process monitoring. The subject covers a vast and diverse range of works, which are compiled and critically evaluated based on the different perspectives they provide. Data-driven modeling techniques are surveyed and categorized into two main groups: multivariate statistics and machine learning. Representative models, namely principal component analysis, partial least squares and artificial neural networks, are detailed in a didactic manner. Topics not typically covered by other reviews, such as process data exploration and treatment, software and benchmarks availability, and real-world industrial implementations, are thoroughly analyzed. Finally, future research perspectives are discussed, covering aspects related to system performance, the significance and usefulness of the approaches, and the development environment. This work aims to be a reference for practitioners and researchers navigating the extensive literature on data-driven industrial process monitoring.