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As big data becomes ubiquitous across domains, and more and more stakeholders aspire to make the most of their data, demand for machine learning tools has spurred researchers to explore the possibilities of automated machine learning (AutoML). AutoML tools aim to make machine learning accessible for non-machine learning experts (domain experts), to improve the efficiency of machine learning, and to accelerate machine learning research. But although automation and efficiency are among AutoML's main selling points, the process still requires human involvement at a number of vital steps, including understanding the attributes of domain-specific data, defining prediction problems, creating a suitable training data set, and selecting a promising machine learning technique. These steps often require a prolonged back-and-forth that makes this process inefficient for domain experts and data scientists alike, and keeps so-called AutoML systems from being truly automatic. In this review article, we introduce a new classification system for AutoML systems, using a seven-tiered schematic to distinguish these systems based on their level of autonomy. We begin by describing what an end-to-end machine learning pipeline actually looks like, and which subtasks of the machine learning pipeline have been automated so far. We highlight those subtasks which are still done manually - generally by a data scientist - and explain how this limits domain experts' access to machine learning. Next, we introduce our novel level-based taxonomy for AutoML systems and define each level according to the scope of automation support provided. Finally, we lay out a roadmap for the future, pinpointing the research required to further automate the end-to-end machine learning pipeline and discussing important challenges that stand in the way of this ambitious goal.