The physical state of a robotic system naturally carries structure; the pose of rigid links can be written as elements of the Special Euclidean group, images taken by a camera of a planar scene can be related by homographies and mapped to elements of the special linear group, etc. Recent work has demonstrated that there is a rich collection of symmetry groups for different robotic problems above and beyond the classical Lie-groups. This talks shows how this structure can be exploited to design robust nonlinear observers for state estimation. The earliest results in this direction were nonlinear attitude estimators (2005-2010) that were an enabling technology in the aerial robotic vehicle industry. Pose estimation algorithms based on these ideas are built into the augmented reality headsets that are now ubiquitous in gaming. Recent symmetries have opened the door to new solutions for classical robotics problems such as visual odometry, visual inertial odometry, simultaneous localisation and mapping.