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Survey telescopes such as the Vera C. Rubin Observatory and the Square Kilometre Array will discover billions of static and dynamic astronomical sources. Properly mined, these enormous datasets will likely be wellsprings of rare or unknown astrophysical phenomena. The challenge is that the datasets are so large that most data will never be seen by human eyes; currently the most robust instrument we have to detect relevant anomalies. Machine learning is a useful tool for anomaly detection in this regime. However, it struggles to distinguish between interesting anomalies and irrelevant data such as instrumental artefacts or rare astronomical sources that are simply not of interest to a particular scientist. Active learning combines the flexibility and intuition of the human brain with the raw processing power of machine learning. By strategically choosing specific objects for expert labelling, it minimises the amount of data that scientists have to look through while maximising potential scientific return. Here we introduce Astronomaly: a general anomaly detection framework with a novel active learning approach designed to provide personalised recommendations. Astronomaly can operate on most types of astronomical data, including images, light curves and spectra. We use the Galaxy Zoo dataset to demonstrate the effectiveness of Astronomaly, as well as simulated data to thoroughly test our new active learning approach. We find that for both datasets, Astronomaly roughly doubles the number of interesting anomalies found in the first 100 objects viewed by the user. Astronomaly is easily extendable to include new feature extraction techniques, anomaly detection algorithms and even different active learning approaches. The code is publicly available at https://github.com/MichelleLochner/astronomaly.