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.)