We study general M-estimators of location on Riemannian manifolds, extending classical notions such as the Frechet mean by replacing the squared loss with a broad class of loss functions. Under minimal regularity conditions on the loss function and the underlying probability distribution, we establish theoretical guarantees for the existence and uniqueness of such estimators. In particular, we provide sufficient conditions under which the population and sample M-estimators exist and are uniquely defined. Our results offer a general framework for robust location estimation in non-Euclidean geometric spaces and unify prior uniqueness results under a broad class of convex losses.