Summary: We prove some symmetry theorems for positive solutions of elliptic equations in some noncompact manifolds, which generalize and extend symmetry results known in the case of the euclidean space \(\mathbb{R}^n\). The (variational) technique that we use relies on Sobolev inequalities available for manifolds together with the well-known method of moving planes. In the particular case of the standard \(n\)-dimensional hyperbolic space \(\mathbb{H}^n\) we get the radial symmetry of positive solutions of the equation \(-\Delta_{\mathbb{H}^n} u=f(u)\) in \(\mathbb{H}^n\), which tend to zero at infinity (or belong to the Sobolev space \(H^1(\mathbb{H}^n)\) in some cases), under different hypotheses on the relationship between the behavior of the nonlinearity \(f\) in a neighborhood of zero and the summability properties of the solution. One of the main features of this work is to single out and study the connection between the geometric properties of the manifold considered and the growth conditions on the nonlinearity in order to have our symmetry results.