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We introduce the Automatic Learning for the Rapid Classification of Events (ALeRCE) broker, an astronomical alert broker designed to provide a rapid and self--consistent classification of large etendue telescope alert streams, such as that provided by the Zwicky Transient Facility (ZTF) and, in the future, the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST). ALeRCE is a Chilean--led broker run by an interdisciplinary team of astronomers and engineers, working to become intermediaries between survey and follow--up facilities. ALeRCE uses a pipeline which includes the real--time ingestion, aggregation, cross--matching, machine learning (ML) classification, and visualization of the ZTF alert stream. We use two classifiers: a stamp--based classifier, designed for rapid classification, and a light--curve--based classifier, which uses the multi--band flux evolution to achieve a more refined classification. We describe in detail our pipeline, data products, tools and services, which are made public for the community (see \url{https://alerce.science}). Since we began operating our real--time ML classification of the ZTF alert stream in early 2019, we have grown a large community of active users around the globe. We describe our results to date, including the real--time processing of $9.7\times10^7$ alerts, the stamp classification of $1.9\times10^7$ objects, the light curve classification of $8.5\times10^5$ objects, the report of 3088 supernova candidates, and different experiments using LSST-like alert streams. Finally, we discuss the challenges ahead to go from a single-stream of alerts such as ZTF to a multi--stream ecosystem dominated by LSST.