In this talk, we present recent results about the boundedness in $L^p$ spaces for all $1<p<\infty$ of the so-called vertical Littlewood--Paley functions for non-local Dirichlet forms in the metric measure space. For $1<p\le 2$, the pseudo-gradient is used to overcome the difficulty that chain rules are not valid for non-local operators, and the Mosco convergence paves the way from finite jumping kernel case to general case, while for $2\le p<\infty$, the Burkholder--Gundy inequality is effectively applied. The former method is analytic and the latter one is probabilistic. The results extend those ones for pure jump symmetric L\'evy processes in Euclidean spaces.