  For an arc-colored digraph $D$, define its {\em kernel by rainbow paths} to
be a set $S$ of vertices such that (i) no two vertices of $S$ are connected by
a rainbow path in $D$, and (ii) every vertex outside $S$ can reach $S$ by a
rainbow path in $D$. In this paper, we show that it is NP-complete to decide
whether an arc-colored tournament has a kernel by rainbow paths, where a {\em
tournament} is an orientation of a complete graph. In addition, we show that
every arc-colored $n$-vertex tournament with all its strongly connected
$k$-vertex subtournaments, $3\leq k\leq n$, colored with at least $k-1$ colors
has a kernel by rainbow paths, and the number of colors required cannot be
reduced.
