Abstract: Let $\Delta$ be a building and let $\Gamma$ be a subgroup of ${\rm Aut}(\Delta)$. After presenting some basic notions, we will describe a simple necessary and sufficient condition for the set $\Delta^\Gamma$ of residues of $\Delta$ stabilized by $\Gamma$ to form a building. When this condition is satisfied, we say that $\Gamma$ is a descent group of $\Delta$. To each descent group $\Gamma$ there is an associated Tits index that captures the key features of both the building $\Delta^\Gamma$ and the action of $\Gamma$. We will describe several applications of our results involving both spherical and affine buildings. This is joint work with Bernhard M\"uhlherr and Holger Petersson.