The author constructs a probabilistic version of Nevanlinna theory in general cases, improving the stochastic method by estimates of increasing processes for Brownian motions (Br.m.) and martingales (mart.) on manifolds. In Section 1 he establishes counterparts of the 1. and 2. Main Nevanlinna Theorems and of a defect relation in the general situation of a filtered probability space \((\Omega, {\mathcal F}, P, ({\mathcal F}_t)_{t\geq 0})\) for a nonnegative continuous \({\mathcal F}_t\)-local submart. \((V_t) {t\geq 0}\) on it, after decomposing \(V_t= M_t+ A_t\) with \(M_t\) a local mart. and \(A_t\) an increasing process. Thus an analogy of the 1. Theorem corresponds to \[ E[V_T]- E[V_0]+ N(T,V)= E[A_T], \] where \(T\) is an \({\mathcal F}_t\)-stopping time with \(E[A_T] \lambda)\), \(V^*_T= \sup\{V_t: 0\leq t\leq T\}\), and the 2. Theorem to \[ E[V_T] \leq-E [\log\rho_T] +\log E \biggl[ \exp \bigl(V_T+ h(V_T) \bigr)\rho_T\biggr] +h\bigl(E[V_T] \bigr), \] where \(\rho_t\) is a positive \({\mathcal F}_t\)-adapted process and \(h\) a convex function on \([0,\infty)\). Further taking a sequence of stopping times \(T_n\), and \(h\) such that convenient inequalities hold, a defect relation is obtained in the form \(\liminf_{n\to\infty} (E[V_{T_n}]/E) \leq\limsup_{n\to \infty} (-E [\log \rho_{T_n}]/E )\), \(\mathbb{E}= E[\int^{T_n}_0 \rho_s ds]\). Section 2 contains a main estimate relative to a Green function and a harmonic measure of a Br.m. on manifolds. The author considers two types of \(n\)-dimensional Riemannian manifolds \(M\): the Cartan-Hadamard in sense of Greene-Wu, in particular those which are also spherically symmetric, and the parabolic ones defined by an exhaustion function. He proves estimates for the image of a Br.m. \(X_t\) on \(M\) under a positive continuous function \(f\): \[ E \bigl[\log f(X_{T_r}) \bigr] \leq (\beta+1)^2 \log E \left[\int^{T_r}_0 f(X_s)ds \right]+ S(r), \] giving the expression of \(T_r\) and \(S(r)\) in all the three cases, \(\beta>0\), \(r\in (0,\infty) \smallsetminus E_\beta\), \(E_\beta\) of finite Lebesgue measure. Section 3 is dedicated to harmonic morphisms \(\Phi: M\to N\) between Riemannian \(n\)-manifolds \(M\) and \(N\), which are characterized (Darling, Fuglede) as mappings a Br.m. \(X_t\) on \(M\) to a diffusion process \(Y_t\) on \(N\), \(Y_{A_t} =\Phi (X_t)\) with \(A_t= \int^t_0 \lambda^2_\Phi (X_s) ds\), \(\lambda_\Phi\)= the dilatation of \(\Phi\). For \(M\) as above, \(u\) a nonnegative Borel measurable function on \(N\) such that \(u(Y_t)\) be a continuous local submart., and from the inequality \(\sup_x E_x [\int^1_0 (\exp (u+h(u)) (Y_s)ds] <\infty\), an analogy of the 2. Theorem is deduced. (Procedure works if the image is a general Markov process too). The results are illustrated for \(N\) a compact Riemannian manifold with strictly positive injective radius at every point, by constructing a function \(u_S\), \(S\) being a family of \(q\) disjoint smooth submanifolds in \(N\) of codimension 2. Again Nevanlinna type 1 and 2 theorems and a defect relation are given. The conformal case \(M=\) Riemann surface, \(N= \mathbb{P}^1\), \(\Phi\) a meromorphic function clarifies the relation to classical Nevanlinna theory, \(M = \mathbb{C}\) or the unit disc, as well as to the case \(M=a\) parabolic Riemann surface. Section 4 treats the case when the image of the Br.m. is a mart.: \(Y_t\) on a Riemannian manifold \(N\) with Levi-Civita connexion. New concepts are associated to \(Y_t\) and estimates given using Krylov's one. For \(M\) as in 2., \(N\) compact, \(u\) and \(h\) satisfying some conditions, a 2. type theorem is deduced if \(\Phi\) is a stochastic nondegenerate harmonic map. Then the holomorphic case is dealt with and conformal martingales on Kähler manifolds are considered. A 2. type theorem is proved for \(\Phi\) a nondegenerate holomorphic map: \(\mathbb{C}^n \to\mathbb{P}^n\), with interesting consequences on defect relations, in particular a relation on the defect of an algebraic hypersurface which improves a Carlson-Griffiths' one. The theorem extends to a complex projective algebraic \(n\)-manifold \(N\). Section 5 contains an averaging Bezout estimate via probabilistic methods, for a harmonic morphism \(\Phi\), which extends to maps preserving strong Markov processes up to time-change.