10. Let $x$ and $y$ be real numbers, and satisfy $\left\{\begin{array}{l}(x-1)^{3}+2003(x-1)=-1 \\ (y-1)^{3}+2003(y-1)=1,\end{array} x+y=\right.$
10. 2.
$\left\{\begin{array}{l}(x-1)^{3}+2003(x-1)=-1 \\ (1-y)^{3}+2003(1-y)=-1\end{array}\right.$ Since $f(t)=t^{3}+2003 t$ is monotonically increasing on $(-\infty,+\infty)$, and $f(x-1)=f(1-y)$, it follows that $x-1=1-y$, thus $x+y=2$.