  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 $\pi_n(X)\ne 0$ 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.
