Climate change is global, yet its concrete impacts can strongly vary between different locations in the same region. Seasonal weather forecasts currently operate at the mesoscale (> 1 km). For more targeted mitigation and adaptation, modelling impacts to < 100 m is needed. Yet, the relationship between driving variables and Earth's surface at such local scales remains unresolved by current physical models. Large Earth observation datasets now enable us to create machine learning models capable of translating coarse weather information into high-resolution Earth surface forecasts. Here, we define high-resolution Earth surface forecasting as video prediction of satellite imagery conditional on mesoscale weather forecasts. Video prediction has been tackled with deep learning models. Developing such models requires analysis-ready datasets. We introduce EarthNet2021, a new, curated dataset containing target spatio-temporal Sentinel 2 satellite imagery at 20 m resolution, matched with high-resolution topography and mesoscale (1.28 km) weather variables. With over 32000 samples it is suitable for training deep neural networks. Comparing multiple Earth surface forecasts is not trivial. Hence, we define the EarthNetScore, a novel ranking criterion for models forecasting Earth surface reflectance. For model intercomparison we frame EarthNet2021 as a challenge with four tracks based on different test sets. These allow evaluation of model validity and robustness as well as model applicability to extreme events and the complete annual vegetation cycle. In addition to forecasting directly observable weather impacts through satellite-derived vegetation indices, capable Earth surface models will enable downstream applications such as crop yield prediction, forest health assessments, coastline management, or biodiversity monitoring. Find data, code, and how to participate at www.earthnet.tech .