10. $A B C D-A_{1} B_{1} C_{1} D_{1}$ is a cube with edge length 1, $D 、 E 、 F 、 G$ are the midpoints of $B A$ 、 $B C 、 B B_{1}$ and $A_{1} D_{1}$, respectively, then the volume of the tetrahedron $G-D E F$ is
10. $\frac{1}{12}$

Take the midpoints $H, \mathrm{I}$ of $A A_{1}$ and $A D$.
It is easy to prove that $D H / / E G$, so $D H / /$ plane $E F G$.
Therefore, $V_{D-E P G}=V_{H-E F G}$.
It is easy to prove that $E / / H F$ so $I E / /$ plane $F G H$.
Thus, $V_{E-H K_{3}}=V_{I-Y Y H}$.
Therefore, $V_{G-D F}=V_{F-G H}=\frac{1}{3} \cdot S_{\triangle G H} \cdot H F=\frac{1}{3} \cdot \frac{1}{4} \cdot 1=\frac{1}{12}$.