  A main objective of the present paper is to develop the theory of
hypercyclicity and supercyclicity of linear operators on topological vector
space over non-Archimedean valued fields. We show that there does not exist any
hypercyclic operator on finite dimensional spaces. Moreover, we give sufficient
and necessary conditions of hypercyclicity (resp. supercyclicity) of linear
operators on separable $F$-spaces. It is proven that a linear operator $T$ on
topological vector space $X$ is hypercyclic (supercyclic) if it satisfies
Hypercyclic (resp. Supercyclic) Criterion. We consider backward shifts on
$c_0$, and characterize hypercyclicity and supercyclicity of such kinds of
shifts. Finally, we study hypercyclicity, supercyclicity of operators $\lambda
I+\mu B$, where $I$ is identity and $B$ is backward shift. We note that there
are essential differences between the non-Archimedean and real cases.
