  In this article, we study the magnitude homology of geodesic metric spaces of
curvature $\leq \kappa$, especially ${\rm CAT}(\kappa)$ spaces. We will show
that the magnitude homology $MH^{l}_{n}(X)$ of such a meric space $X$ vanishes
for small $l$ and all $n > 0$. Conseqently, we can compute a total
$\mathbb{Z}$-degree magnitude homology for small $l$ for the shperes
$\mathbb{S}^{n}$, the Euclid spaces $\mathbb{E}^{n}$, the hyperbolic spaces
$\mathbb{H}^{n}$, and real projective spaces $\mathbb{RP}^{n}$ with the
standard metric. We also show that an existence of closed geodesic in a metric
space guarantees the non-triviality of magnitude homology.
