Periodic geodesics in singular spaces

We extend the classical result of Lyusternik and Fet on the existence of closed geodesics to singular spaces. We show that if $X$ is a compact geodesic metric space satisfying the CAT($\kappa$) condition for some fixed $\kappa >0$ and ${\pi }_n(X)\mathrm{\ne }0$ for some $n>0$ then $X$ has a periodic geodesic. This condition is satisfied for example by locally CAT($\kappa$) manifolds. Our result applies more generally to compact locally uniquely geodesic spaces.