Modern approach to artificial intelligence (AI) aims to design algorithms that learn directly from data. This approach has achieved impressive results and has contributed significantly to the progress of AI, particularly in the sphere of supervised deep learning. It has also simplified the design of machine learning systems as the learning process is highly automated. However, not all data processing tasks in conventional deep learning pipelines have been automated. In most cases data has to be manually collected, preprocessed and further extended through data augmentation before they can be effective for training. Recently, special techniques for automating these tasks have emerged. The automation of data processing tasks is driven by the need to utilize large volumes of complex, heterogeneous data for machine learning and big data applications. Today, end-to-end automated data processing systems based on automated machine learning (AutoML) techniques are capable of taking raw data and transforming them into useful features for Big Data tasks by automating all intermediate processing stages. In this work, we present a thorough review of approaches for automating data processing tasks in deep learning pipelines, including automated data preprocessing--e.g., data cleaning, labeling, missing data imputation, and categorical data encoding--as well as data augmentation (including synthetic data generation using generative AI methods) and feature engineering--specifically, automated feature extraction, feature construction and feature selection. In addition to automating specific data processing tasks, we discuss the use of AutoML methods and tools to simultaneously optimize all stages of the machine learning pipeline.