  In this paper, we establish a type of uniqueness and nondegeneracy results
for positive solutions to the following nonlocal Kirchhoff equations
\begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\text{d}
x\right)\Delta u+u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3}, \end{eqnarray*}
where $a,b$ are positive constants and $1<p<5$. Before this paper, it seems
that there have no this type of results even on positive ground states
solutions to Kirchhoff type equations, much less on general positive solutions.
To overcome the difficulty brought by the nonlocality, some new observation on
Kirchhoff equations is found, and some related theories on classical
Schr\"odinger equations are applied.
