Kaplansky asked about the set $S$ of possible images of a polynomial $f$ in several noncommuting variables in the matrix algebra $M_n(F)$ over a field $F$. It follows from work of Herstein that the space spanned by $S$ must either be scalar or contain $sl_n$. After a review of our earlier work for $n \leq 3$, when $K$ is closed under quadratic extensions, we turn to the case of a Lie polynomial with constant term $0$, and coefficients in an algebraically closed field $K$. We describe all the possible images of $f$ in $M_2(K)$. An example is given of a polynomial $f$ whose image is the set of trace zero matrices excluding nilpotent nonzero matrices, together with an arithmetic criterion for this case. The Lie case has special interest since many celebrated polynomials such as the standard polynomial are not Lie polynomials. Some Lie results are provided for $n = 3$, together with an indication of what remains open.
(Joint work with Kanel-Belov and Malev.)