This dissertation contains several new results concerning Moser-type optimal stopping problems. In the simplest case we consider sequence of independent uniformly distributed points $X_{1}, X_{2},\cdots,X_{n}$ on the compact Riemannian manifold $\mathcal{M}$ and give algorithm for the calculation of $S_{n}=\max\limits_{\tau\leq n}E[\mathcal{G}(X_{\tau})]$ where $\mathcal{G}$ is a smooth function on $\mathcal{M}$ and $\tau$ is a random optimal stopping time. Description of the optimal $\tau$ depends on the structure of $\mathcal{G}$ near points of maximum. For different assumptions on this structure we calculate asymptotics of $S_{n}$.