Totally geodesic submanifolds play a fundamental role in Riemannian geometry and they are closely related to symmetry. For this reason, their study is particularly rich in highly symmetric settings, such as homogeneous and symmetric spaces, providing deep insights into their underlying geometric structures and properties. In this talk, I will present a broad overview of recent developments in the classification, construction, and applications of totally geodesic submanifolds in these spaces. Emphasizing both classical results and new advances, I will discuss their connections to Lie group theory.