The main interest of my research is geometric function theory with stochastic calculus. It is well-known that the composition of Brownian motion and a harmonic function is a martingale. It is also well-known that the composition of complex Brownian motion and a holomorphic function is a conformal martinagle. These relationships can be extended to higher dimensional cases with manifold valued martingales and holomorphic martingales. We can obtain function theoretic results by studying such stochastic processes. The focus of my research is the global behavior of Brownian motion and martingales on Riemaniann manifolds, properties of harmonic functions and harmonic maps, function theoretic and probabilistic properties of minimal submanifolds, Nevanlinna theory on submanifolds and Kähler manifolds.