Sufficient conditions are obtained for the continuity of renormalized self-intersection local times for the multiple intersections of a large class of strongly symmetric L vy processes in $R^m$, $m=1,2$. In $R^2$ these include Brownian motion and stable processes of index greater than 3/2, as well as many processes in their domains of attraction. In $R^1$ these include stable processes of index $3/4<\beta\le 1$ and many processes in their domains of attraction.