Mechanical engineers use submanifolds of the special Euclidean group SE(3) to characterize the motion pattern of a mechanism or robot manipulator. So far, Lie subgroups of SE(3) are well known to the mechanism community, and their extended application in mechanism analysis and synthesis are well documented in the literature. General submanifolds, on the other hand, are hard to classify and not all of them are useful for real applications. Recently, we discovered that there is a special class of submanifolds of SE(3) that have potentially broad application in mechanism analysis and synthesis, yet are largely overlooked for forty years. These are the symmetric subspaces (or totally geodesic submanifolds) of SE(3), with SE(3) treated as a sym- metric space. In this seminar, I will give an informal introduction of these submanifolds from an mechanical engineering point of view. First, I will briefly review the state-of-the-art application of Lie subgroups of SE(3) in mechanism analysis and synthesis. The limitation of Lie subgroups is illustrated by some motivating engineering examples. Second, I will give a brief introduction to symmetric subspaces of SE(3) using Ottmar Loos’ elementary approach. I will study properties of these submanifolds and show how they can be used to design new mechanisms.