With the rise of deep learning, there has been renewed interest within the process industries to utilize data on large-scale nonlinear sensing and control problems. We identify key statistical and machine learning techniques that have seen practical success in the process industries. To do so, we start with hybrid modeling to provide a methodological framework underlying core application areas: soft sensing, process optimization, and control. Soft sensing contains a wealth of industrial applications of statistical and machine learning methods. We quantitatively identify research trends, allowing insight into the most successful techniques in practice. We consider two distinct flavors for data-driven optimization and control: hybrid modeling in conjunction with mathematical programming techniques and reinforcement learning. Throughout these application areas, we discuss their respective industrial requirements and challenges. A common challenge is the interpretability and efficiency of purely data-driven methods. This suggests a need to carefully balance deep learning techniques with domain knowledge. As a result, we highlight ways prior knowledge may be integrated into industrial machine learning applications. The treatment of methods, problems, and applications presented here is poised to inform and inspire practitioners and researchers to develop impactful data-driven sensing, optimization, and control solutions in the process industries.