Data-driven three-dimensional parametric shape models of the human body have gained enormous popularity both for the analysis of visual data and for the generation of synthetic humans. Following a similar approach for animals does not scale to the multitude of existing animal species, not to mention the difficulty of accessing subjects to scan in 3D. However, we argue that for domestic species of great importance, like the horse, it is a highly valuable investment to put effort into gathering a large dataset of real 3D scans, and learn a realistic 3D articulated shape model. We introduce VAREN, a novel 3D articulated parametric shape model learned from 3D scans of many real horses. VAREN bridges synthesis and analysis tasks, as the generated model instances have unprecedented realism, while being able to represent horses of different sizes and shapes. Differently from previous body models, VAREN has two resolutions, an anatomical skeleton, and interpretable, learned pose-dependent deformations, which are related to the body muscles. We show with experiments that this formulation has superior performance with respect to previous strategies for modeling pose-dependent deformations in the human body case, while also being more compact and allowing an analysis of the relationship between articulation and muscle deformation during articulated motion. The VAREN model and data are available at https://varen.is.tue.mpg.de.