Title: Geomstats: a Python Package for Riemannian Geometry in Machine Learning. Abstract: We introduce geomstats, a Python package for Riemannian modelization and optimization over manifolds such as hyperspheres, hyperbolic spaces, SPD matrices or Lie groups of transformations. Our contribution is threefold. First, geomstats allows the flexible modeling of many a machine learning problem through an efficient and extensively unit-tested implementations of these manifolds, as well as the set of useful Riemannian metrics, exponential and logarithm maps that we provide. Moreover, the wide choice of loss functions and our implementation of the corresponding gradients allow fast and easy optimization over manifolds. Finally, geomstats is the only package to provide a unified framework for Riemannian geometry, as the operations implemented in geomstats are available with different computing backends (numpy,tensorflow and keras), as well as with a GPU-enabled mode–-thus considerably facilitating the application of Riemannian geometry in machine learning. In this paper, we present geomstats through a review of the utility and advantages of manifolds in machine learning, using the concrete examples that they span to show the efficiency and practicality of their implementation using our package