GaBO is a Geometry-aware Bayesian Optimization framework that exploits the geometry of non-Euclidean parameter spaces, which often arise in robotics (e.g. orientation, stiffness matrix, manipulability ellipsoids, inertia tensors). GaBO, built on Riemannian manifold theory, allows Bayesian optimization to properly measure similarities in the parameter space through geometry-aware kernel functions and to optimize the acquisition function on the manifold as an unconstrained problem. We test our approach in several benchmark artificial landscapes and using a 7-DOF simulated robotic manipulator to orientation and impedance parameters.