  In this paper we study the quantization problem for probability measures on
Riemannian manifolds. Under a suitable assumption on the growth at infinity of
the measure we find asymptotic estimates for the quantization error,
generalizing the results on $\mathbb{R}^d.$ Our growth assumption depends on
the curvature of the manifold and reduces, in the flat case, to a moment
condition. We also build an example showing that our hypothesis is sharp.
