  Symmetric spaces arise in wide variety of problems in Mathematics and
Physics. They are mostly studied in Representation theory, Harmonic analysis
and Differential geometry. As many physical systems have symmetric spaces as
their configuration spaces, the study of controllability on symmetric space is
quite interesting. In this paper, a driftless control system of type \dot{x}=
\sum_{i=1}^m u_if_i(x) is considered on a symmetric space. For this we have
established global controllability condition which is illustrated by few
examples of exponential submanifolds of SE(3) and random matrix ensembles.
