  For a fixed graph $F$, we would like to determine the maximum number of edges
in a properly edge-colored graph on $n$ vertices which does not contain a
{\emph rainbow copy} of $F$, that is, a copy of $F$ all of whose edges receive
a different color. This maximum, denoted by $ex^*(n,F)$, is the {\emph rainbow
Tur\'an number} of $F$, and its systematic study was initiated by Keevash,
Mubayi, Sudakov and Verstra\"ete in 2007. We determine $ex^*(n,F)$ exactly when
$F$ is a forest of stars, and give bounds on $ex^*(n,F)$ when $F$ is a path
with $k$ edges, disproving a conjecture in Keevash et al.
