in this talk, we will discuss (a certain form of) the siegel-weil formula for the second terms (the weak second term identity). as an application, we show the following non-vanishing result of global theta lifts from orthogonal groups: let $\pi$ be a cuspidal automorphic representation of an orthogonal group $o(v)$ with $\dim v=2r-j$ even and $0\leq j\leq r-1$. then the global theta lift of $\pi$ to $sp(2r)$ does not vanish ''up to disconnectedness" if the (incomplete) $l$-function $l^s(s,\pi)$ does not vanish at $s=1+\frac{j}{2}$. (this is a joint with w. gan.)