  Let $\frak a$ denote an ideal in a commutative Noetherian ring $R$ and $M$ a
finitely generated $R$-module. In this paper, we introduce the concept of the
cohomological dimension filtration $\mathscr{M} =\{M_i\}_{i=0}^c$, where $
c={\rm cd} ({\frak a},M)$ and $M_i$ denotes the largest submodule of $M$ such
that ${\rm cd} ({\frak a}, M_i)\leq i.$ Some properties of this filtration are
investigated. In particular, in the case that $(R, \frak m)$ is local and $c=
\dim M$, we are able to determine the annihilator of the top local cohomology
module $H_{\frak a}^c(M)$. In fact, it is shown that ${\rm Ann}_R(H_{\frak
a}^c(M))= {\rm Ann}_R(M/M_{c-1}).$ As a consequence, it follows that there
exists an ideal $\frak b$ of $R$ such that ${\rm Ann}_R(H_{\frak
a}^{c}(M))={\rm Ann}_R(M/H_{\frak b}^{0}(M))$.
