Let \(0<q\leq p<\infty\), and \(B_R(x)\) be the open ball centered at \(x\) and of radius \(R\). The Morrey space \(M^p_q(\mathbb R^n)\) is defined to be the set of locally \(q\)-integrable functions \(f\) on \(\mathbb R^n\) such that \[ \| f\| _{M^p_q(\mathbb R^n)}=\sup_{x\in{\mathbb R^n},\, R>0} R^{n(1/p-1/q)}\left(\int_{B_R(x)}| f(y)| ^q\,dy\right)^{1/q}<\infty. \] Let \(\phi=\{\phi_j\}_{j=0}^\infty\subset{\mathcal S}(\mathbb R^n)\) be a set of real-valued even functions with certain extra conditions on their supports and the decay of their derivatives, and \(\sum^\infty_{j=0}\phi_j(x)=1\) for all \(x\in\mathbb R^n\). Let \(s\in\mathbb R\), \(0<q\leq p<\infty\) and \(0<\beta\leq\infty\). The Morrey-type Besov space \(MB^{s,\beta}_{p,q}(\mathbb R^n)\) is defined to be the set of Schwartz distributions \(f\) such that \[ \| f\| ^\phi_{MB^{s,\beta}_{p,q}(\mathbb R^n)} =\left\{\sum^\infty_{k=0}2^{ks\beta}\| {\mathcal F}^{-1} \phi_j{\mathcal F}f\| _{M^p_q(\mathbb R^n)}^\beta \right\}^{1/\beta}<\infty, \] where \({\mathcal F}\) is the Fourier transform and \({\mathcal F}^{-1}\) is its inverse. The Morrey-type Triebel space \(MF^{s,\beta}_{p,q}(\mathbb R^n)\) is defined to be the set of Schwartz distributions \(f\) such that \[ \| f\| ^\phi_{MF^{s,\beta}_{p,q}(\mathbb R^n)} =\left\| \left\{\sum^\infty_{k=0}2^{ks\beta}| {\mathcal F}^{-1} \phi_j{\mathcal F}f| ^\beta \right\}^{1/\beta}\right\| _{M^p_q(\mathbb R^n)}<\infty. \] (The norm \(\| \cdot\| ^\phi_{MF^{s,\beta}_{p,q}(\mathbb R^n)}\) on page 905 of the paper may not be correct.) Obviously, when \(s\in\mathbb R\), \(0<p=q<\infty\) and \(0<\beta\leq\infty\), these spaces become the classical Besov spaces and Triebel spaces. The authors verify that these spaces are independent of choices of \(\phi\). Then they establish lifting properties, Fourier multiplier theorems and discrete characterizations of these spaces. As an application, they obtain the boundedness of some operators, including pseudo-differential operators of the Hörmander class in these spaces.