Let \(\mathcal{M}(E)\) denote the set of all Lebesgue measurable functions on \(E\). A quasi-Banach space \(X\subset \mathcal{M}(\mathbb{R}^n)\) is called a rearrangement-invariant quasi-Banach function space if there exists a quasi-norm \(\rho_X\) defined on the set of Lebesgue measurable functions on \((0,\infty)\) such that {\parindent=6mm \begin{itemize}\item[(1)] \(\rho_X(f)=0\) if and only if \(f=0\) a.\,e.; \item[(2)] \(|g|\leq |f|\) a.\,e. implies that \(\rho_X(g)\leq \rho_X(f)\); \item[(3)] \(0\leq f_n \uparrow f\) a.\,e. implies that \(\rho_X(f_n)\uparrow \rho_X(f)\); \item[(4)] \(\chi_E\in \mathcal{M}(0,\infty)\) and \(|E|<\infty\) implies that \(\rho_X(\chi_E)<\infty.\) \end{itemize}} For \(f\in X\), let \(\|f\|_X=\rho_X(f^*)\), where \(f^*\) is the non-increasing rearrangement of \(f\). Many classical function spaces, like the Lorentz spaces and the Orlicz spaces, are rearrangement-invariant quasi-Banach function spaces. In this paper, some classical results on the mapping properties of the fractional integral operators, the convolution operators, the Fourier integral operators and the oscillatory operators, the Fourier restriction theorem and the Sobolev embedding theorem, as well as the interpolation theorem of operators, are extended to rearrangement-invariant quasi-Banach function spaces. Based on these results, the author introduces and develops some Triebel-Lizorkin-type and Hardy-type spaces built on rearrangement-invariant quasi-Banach function spaces.