  In this article, we introduce the class
$\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)$ of harmonic $\alpha$-Bloch-type
mappings on $\Omega$ as a generalization of the class
$\mathcal{B}_{\mathcal{H},\Omega}(\alpha)$ of harmonic $\alpha$-Bloch mappings
on $\Omega$, where $\Omega$ is arbitrary proper simply connected domain in the
complex plane. We study several interesting properties of the classes
$\mathcal{B}_{\mathcal{H},\Omega}(\alpha)$ and
$\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)$ on arbitrary proper simply
connected domain $\Omega$ and on the shifted disk $\Omega_{\gamma}$ containing
$\mathbb{D}$, where $$ \Omega_{\gamma}:=\bigg\{z\in\mathbb{C} :
\bigg|z+\frac{\gamma}{1-\gamma}\bigg|<\frac{1}{1-\gamma}\bigg\ $$ and $0 \leq
\gamma <1$. We establish the Landau's theorem for the harmonic Bloch space
$\mathcal{B}_{\mathcal{H},\Omega _{\gamma}}(\alpha)$ on the shifted disk
$\Omega_{\gamma}$. For $f \in \mathcal{B}_{\mathcal{H},\Omega}(\alpha)$
(respectively $\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)$) of the form
$f(z)=h(z) + \overline{g(z)}=\sum_{n=0}^{\infty}a_nz^n +
\overline{\sum_{n=1}^{\infty}b_nz^n}$ in $\mathbb{D}$ with Bloch norm
$||f||_{\mathcal{H},\Omega, \alpha} \leq 1$ (respectively
$||f||^{*}_{\mathcal{H},\Omega, \alpha} \leq 1$), we define the Bloch-Bohr
radius for the space $\mathcal{B}_{\mathcal{H},\Omega}(\alpha)$ (respectively
$\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)$) to be the largest radius
$r_{\Omega,f} \in (0,1)$ such that $\sum_{n=0}^{\infty}(|a_n|+|b_{n}|) r^n\leq
1$ for $r \leq r_{\Omega, \alpha}$ and for all $f \in
\mathcal{B}_{\mathcal{H},\Omega}(\alpha)$ (respectively
$\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)$). We investigate Bloch-Bohr
radius for the classes $\mathcal{B}_{\mathcal{H},\Omega}(\alpha)$ and
$\mathcal{B}^{*}_{\mathcal{H},\Omega}(\alpha)$ on simply connected domain
$\Omega$ containing $\mathbb{D}$.
