Several open problems concerning topologically transitive and
hypercyclic continuous linear operators on Hausdorff locally
convex spaces which are not Fr\'echet spaces are solved. Among
others the following results are presented: (1) There exist
transitive operators on the space $ \varphi $ of all finite
sequences endowed with the finest locally convex topology; it was
known that there is no  hypercyclic operator on $\varphi$. (2) The
space of all test functions for distributions, which is also a
complete direct sum of Fr\'echet spaces, admits hypercyclic
operators. (3) Every separable infinitely dimensional Fr\'echet
space contains a dense hyperplane which admits no transitive
operator.