1. Given $f(x)=x^{2}-2 x$, the set
$$
A=\{x \mid f(f(x))=0\} \text {. }
$$

then the sum of all elements in set $A$ is
$-, 1.4$
Notice, the equation $f(f(x))=0$ transforms to $f\left(x^{2}-2 x\right)=0$,
i.e., $\square$
$$
\begin{array}{l}
\left(x^{2}-2 x\right)^{2}-2\left(x^{2}-2 x\right)=0 \\
\Rightarrow\left(x^{2}-2 x\right)\left(x^{2}-2 x-2\right)=0 \\
\Rightarrow x_{1}=0, x_{2}=2, x_{3}=1-\sqrt{3}, x_{4}=1+\sqrt{3} \\
\Rightarrow A=\{0,2,1-\sqrt{3}, 1+\sqrt{3}\} .
\end{array}
$$

Therefore, the sum of all elements in set $A$ is 4.