  Weak amenability of a weighted group algebra, or a Beurling algebra, is a
long-standing open problem. The commutative case has been extensively
investigated and fully characterized. We study the non-commutative case. Given
a weight function $\omega$ on a locally compact group $G$, we characterize
derivations from $L^1(G,\omega)$ into its dual in terms of certain functions.
Then we show that for a locally compact IN group $G$, if there is a non-zero
continuous group homomorphism $\varphi$: $G\to \mathbb{C}$ such that
$\varphi(x)/\omega(x)\omega(x^{-1})$ is bounded on $G$, then $L^1(G,\omega)$ is
not weakly amenable. Some useful criteria that rule out weak amenability of
$L^1(G,\omega)$ are established. Using them we show that for many polynomial
type weights the weighted Heisenberg group algebra is not weakly amenable,
neither is the weighted $\boldsymbol{ax+b}$ group algebra. We further study
weighted quotient group algebra $L^1(G/H,\hat\omega)$, where $\hat\omega$ is
the canonical weight on $G/H$ induced by $\omega$. We reveal that the kernel of
the canonical homomorphism from $L^1(G,\omega)$ to $L^1(G/H,\hat\omega)$ is
complemented. This allows us to obtain some sufficient conditions under which
$L^1(G/H,\hat\omega)$ inherits weak amenability of $L^1(G,\omega)$. We study
further weak amenability of Beurling algebras of subgroups. In general, weak
amenability of a Beurling algebra does not pass to the Beurling algebra of a
subgroup. However, in some circumstances this inheritance can happen. We also
give an example to show that weak amenability of both $L^1(H,\omega|_H)$ and
$L^1(G/H,\hat\omega)$ does not ensure weak amenability of $L^1(G,\omega)$.
