10. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Then the value of $\sum_{k=1}^{2008}\left[\frac{2008 k}{2009}\right]$ is $\qquad$ .
10.2015028.

For $k=1,2, \cdots, 2008$, since $\frac{2008 k}{2009}$ is not an integer, so,
$$
\begin{array}{l}
{\left[\frac{2008 k}{2009}\right]+\left[\frac{2008(2009-k)}{2009}\right]} \\
=\left[\frac{2008 k}{2009}\right]+\left[2008-\frac{2008 k}{2009}\right]=2007 . \\
\text { Therefore, } \sum_{k=1}^{2008}\left[\frac{2008 k}{2009}\right] \\
=\frac{1}{2} \sum_{k=1}^{2008}\left(\left[\frac{2008 k}{2009}\right]+\left[\frac{2008(2009-k)}{2009}\right]\right) \\
=\frac{1}{2} \times 2008 \times 2007=2015028 .
\end{array}
$$