  Let $\pi$ be an irreducible cuspidal automorphic representation of a
quasi-split unitary group ${\rm U}_{\mathfrak n}$ defined over a number field
$F$. Under the assumption that $\pi$ has a generic global Arthur parameter, we
establish the non-vanishing of the central value of $L$-functions,
$L(\frac{1}{2},\pi\times\chi)$, with a certain automorphic character $\chi$ of
${\rm U}_1$, for the case of ${\mathfrak n}=2,3,4$, and for the general
${\mathfrak n}\geq 5$ by assuming a conjecture on certain refined properties of
global Arthur packets. In consequence, we obtain some simultaneous
non-vanishing results for the central $L$-values by means of the theory of
endoscopy.
