  We show that nontrivial solutions to higher and fractional order equations
with certain nonlinearity are radially symmetric and nonincreasing on geodesic
balls in the hyperbolic space $\mathbb{H}^n$ as well as on the entire space
$\mathbb{H}^n$. Applying the Helgason-Fourier analysis techniques on
$\mathbb{H}^n$, we develop a moving plane approach for integral equations on
$\mathbb{H}^n$. We also establish the symmetry to solutions of certain
equations with singular terms on Euclidean spaces. Moreover, we obtain symmetry
to solutions of some semilinear equations involving fractional order
derivatives.
