  It is known that the $L_p$-curvature of a smooth, strictly convex body in
$\mathbb{R}^{n}$ is constant only for origin-centred balls when $1\neq p>-n$,
and only for balls when $p=1$. If $p=-n$, then the $L_{-n}$-curvature is
constant only for origin-symmetric ellipsoids. We prove `local' and `global'
stability versions of these results. For $p\geq 1$, we prove a global stability
result: if the $L_p$-curvature is almost a constant, then the volume symmetric
difference of $\tilde{K}$ and a translate of the unit ball $B$ is almost zero.
Here $\tilde{K}$ is the dilation of $K$ with the same volume as the unit ball.
For $0\leq p<1$, we prove a similar result in the class of origin-symmetric
bodies in the $L^2$-distance. In addition, for $-n<p<0$, we prove a local
stability result: There is a neighborhood of the unit ball that any smooth,
strictly convex body in this neighborhood with `almost' constant
$L_p$-curvature is `almost' the unit ball. For $p=-n$, we prove a global
stability result in $\mathbb{R}^2$ and a local stability result for $n>2$ in
the Banach-Mazur distance.
