  In this report, we first recall the Poincar\'e's classification theorem for
minimal orientation-preserving homeomorphisms on the circle and the Ghys'
classification theorem for minimal orientation-preserving group actions on the
circle. Then we introduce a classification theorem for a specified class of
topologically transitive orientation-preserving group actions on the circle by
$\mathbb Z^d$. Also, some groups that admit/admit no topologically transitive
actions on the line are determined.
