This is a survey on the relations between asymptotic properties of semi- martingales and, in particular, of Brownian motion on a Riemannian manifold on the one hand and curvature properties of the manifold on the other hand. Following a brief description of real-valued semimartingales and of some essentials of calculus on manifolds, an introduction to semimartingales on manifolds is given, including the notions of \(\Gamma\)-martingales and of \(\Gamma\)-Brownian motion, where \(\Gamma\) is a connection. Another (purely deterministic) section is devoted to geodesics and curvature. After this introductory part some - mostly recent - results on relations between curvature, harmonic functions, and harmonic maps on the one hand and transience and recurrence, zero-one laws, limiting directions, the ``Brownian coupling property'', and other properties of Brownian motion on the other hand are surveyed. A sample: boundedness of the Ricci curvature from below by a quadratic polynomial of the distance function implies stochastic completeness of the manifold (i.e., Brownian motion does not explode); for this result a more or less explicit proof is provided. Conversely if the Ricci curvature decreases to -\(\infty\) too fast and if another condition on geodesics is satisfied (namely if the cut-locus is polar) then Brownian motion explodes. Finally ``other topics'' are mentioned very briefly: Malliavin calculus, stochastic flows, stochastic differential forms, small time asymptotics, Wiener sausages, stochastic Kählerian geometry.