Summary: In this paper, we focus on a class of fractional type integral operators that can be served as extensions of Riesz potential with kernels \[ \begin{aligned} K(x,y)=\frac{\Omega_1(x-A_1 y)}{|x-A_1 y |^{{n}/{q_1}}} \cdots \frac{\Omega_m(x-A_m y)}{|x-A_m y |^{{n}/{q_m}}}, \end{aligned} \] where \(\alpha \in [0,n)\), \(m\geqslant 1\), \(\sum \limits_{i=1}^m\frac{n}{q_i}=n-\alpha\), \(\{A_i\}^m_{i=1}\) are invertible matrixes, \(\Omega_i\) is homogeneous of degree 0 on \(\mathbb{R}^n\) and \(\Omega_i\in L^{p_i}(S^{n-1})\) for some \(p_i\in [1,\infty)\). Under appropriate assumptions, we obtain the weighted \(L^p(\mathbb{R}^n)-L^q(\mathbb{R}^n)\) estimates as well as weighted \(H^p(\mathbb{R}^n)-L^q(\mathbb{R}^n)\) estimates of the commutators for such operators with \(BMO\)-type function when \(\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}\). In addition, we acquire the boundedness of these operators and their commutators with a function in Campanato spaces on Orcliz-Morrey spaces as well as the compactness for such commutators in a special case: \(m=1\) and \(A=I\).