  We establish the Kato-type smoothing property, i.e., global-in-time smoothing
estimates with homogeneous weights, for the Schr\"odinger equation on
Riemannian symmetric spaces of non-compact type and general rank. These form a
rich class of manifolds with nonpositive sectional curvature and exponential
volume growth at infinity, e.g., hyperbolic spaces. We achieve it by proving
the Stein-Weiss inequality and the resolvent estimate of the corresponding
Fourier multiplier, which are of independent interest. Moreover, we extend the
comparison principles to symmetric spaces and deduce different types of
smoothing properties for the wave equation, the Klein-Gordon equation, the
relativistic and general orders Schr\"odinger equations. In particular, we
observe that some smoothing properties, which are known to fail on the
Euclidean plane, hold on the hyperbolic plane.
