  The generalized L'vov-Kaplansky conjecture states that for any
finite-dimensional simple algebra $A$ the image of a multilinear polynomial on
$A$ is a vector space. In this paper we prove it for the algebra of octonions
$\mathbb{O}$ over a field satisfying certain specified conditions (in
particular, we prove it for quadratically closed field and for field
$\mathbb{R}$).
  In fact, we prove that the image set must be either $\{0\}$, $F$, the space
of pure octonions $V$, or $\mathbb{O}$. We discuss possible evaluations of
semihomogeneous polynomials on $\mathbb{O}$ and of arbitrary polynomials on the
corresponding Malcev algebra.
