  The paper puts forward new Besov spaces of variable smoothness
$B^{\varphi_{0}}_{p,q}(G,\{t_{k}\})$ and
$\widetilde{B}^{l}_{p,q,r}(\Omega,\{t_{k}\})$ on rough domains. A~domain~$G$ is
either a~bounded Lipschitz domain in~$\mathbb{R}^{n}$ or the epigraph of
a~Lipschitz function, a~domain~$\Omega$ is an $(\varepsilon,\delta)$-domain.
These spaces are shown to be the traces of the spaces
$B^{\varphi_{0}}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$ and
$\widetilde{B}^{l}_{p,q,r}(\mathbb{R}^{n},\{t_{k}\})$ on domains $G$
and~$\Omega$, respectively. The extension operator
$\operatorname{Ext}_{1}:B^{\varphi_{0}}_{p,q}(G,\{t_{k}\}) \to
B^{\varphi_{0}}_{p,q}(\mathbb{R}^{n},\{t_{k}\})$ is linear, the operator
$\operatorname{Ext}_{2}:\widetilde{B}^{l}_{p,q,r}(\Omega,\{t_{k}\}) \to
\widetilde{B}^{l}_{p,q,r}(\mathbb{R}^{n},\{t_{k}\})$ is nonlinear. As
a~corollary, an exact description of the traces of 2-microlocal Besov-type
spaces and weighted Besov-type spaces on rough domains is obtained.
