For a locally symmetric space $M$, we define a compactification $M\cup M\left(\infty \right)$ which we call the “geodesic compactification”. It is constructed by adding limit points in $M\left(\infty \right)$ to certain geodesics in $M$. The geodesic compactification arises in other contexts. Two general constructions of Gromov for an ideal boundary of a Riemannian manifold give $M\left(\infty \right)$ for locally symmetric spaces. Moreover, $M\left(\infty \right)$ has a natural group theoretic construction using the Tits building. The geodesic compactification plays two fundamental roles in...