Understanding cities as complex systems, sustainable urban planning depends on reliable high-resolution data, for example of the building stock to upscale region-wide retrofit policies. For some cities and regions, these data exist in detailed 3D models based on real-world measurements. However, they are still expensive to build and maintain, a significant challenge, especially for small and medium-sized cities that are home to the majority of the European population. New methods are needed to estimate relevant building stock characteristics reliably and cost-effectively. Here, we present a machine learning based method for predicting building heights, which is based only on open-access geospatial data on urban form, such as building footprints and street networks. The method allows to predict building heights for regions where no dedicated 3D models exist currently. We train our model using building data from four European countries (France, Italy, the Netherlands, and Germany) and find that the morphology of the urban fabric surrounding a given building is highly predictive of the height of the building. A test on the German state of Brandenburg shows that our model predicts building heights with an average error well below the typical floor height (about 2.5 m), without having access to training data from Germany. Furthermore, we show that even a small amount of local height data obtained by citizens substantially improves the prediction accuracy. Our results illustrate the possibility of predicting missing data on urban infrastructure; they also underline the value of open government data and volunteered geographic information for scientific applications, such as contextual but scalable strategies to mitigate climate change. ; publishedVersion