To be deployed in our everyday life, robots are required to navigate and act in unstructured and dynamic environments, to react to unknown and uncertain situations, and to integrate the corresponding outcome to their own experience. This entails having not only reliable and stable controllers, but also outstanding adaptation capabilities, so that robot actions lead to successful performance. A key component in both data-driven learning and adaptation is how robots may exploit explicit (e.g. domain knowledge) or implicit (e.g. learned) structures arising in the collected data. 

Domain knowledge and data structures in robotics can be viewed through the lens of geometry: indeed, different variables have specific geometric characteristics, collected data may lie on curved spaces, and various problems can be naturally formulated from a geometric perspective. In this context, differential geometry, or more specifically Lie groups and Riemannian manifold theories provide appropriate methods to cope with the geometry of non-Euclidean spaces. Although geometric methods have been successfully applied to robotics from early on, they only gained interest recently in robot learning, control and optimization. Recent works have shown that bringing geometry-awareness into such robotic problems leads to increased performance, data-efficient learning, and offers strong stability guarantees. 

The main objective of this tutorial is to attract the interest of the robotic community on geometric methods, which have been overlooked in robot learning, control and optimization. Moreover, we expect this tutorial to raise awareness on the importance of differential geometry in the different research branches of robotics. Finally, through a general introduction on geometric methods, as well as an overview of the recent related works in robotics, we aim at providing researchers with the basic tools for integrating geometry into their work.