  Let $F$ be an algebraically closed field of characteristic zero. We consider
the question which subsets of $M_n(F)$ can be images of noncommutative
polynomials. We prove that a noncommutative polynomial $f$ has only finitely
many similarity orbits modulo nonzero scalar multiplication in its image if and
only if $f$ is power-central. The union of the zero matrix and a standard open
set closed under conjugation by $GL_n(F)$ and nonzero scalar multiplication is
shown to be the image of a noncommutative polynomial. We investigate the
density of the images with respect to the Zariski topology. We also answer
Lvov's conjecture for multilinear Lie polynomials of degree at most 4
affirmatively.
