  Let $X$ be a ball quasi-Banach function space satisfying some minor
assumptions. In this article, the authors establish the characterizations of
$H_X(\mathbb{R}^n)$, the Hardy space associated with $X$, via the
Littlewood--Paley $g$-functions and $g_\lambda^\ast$-functions. Moreover, the
authors obtain the boundedness of Calder\'on--Zygmund operators on
$H_X(\mathbb{R}^n)$. For the local Hardy-type space $h_X(\mathbb{R}^n)$
associated with $X$, the authors also obtain the boundedness of
$S^0_{1,0}(\mathbb{R}^n)$ pseudo-differential operators on $h_X(\mathbb{R}^n)$
via first establishing the atomic characterization of $h_X(\mathbb{R}^n)$.
Furthermore, the characterizations of $h_X(\mathbb{R}^n)$ by means of local
molecules and local Littlewood--Paley functions are also given. The results
obtained in this article have a wide range of generality and can be applied to
the classical Hardy space, the weighted Hardy space, the Herz--Hardy space, the
Lorentz--Hardy space, the Morrey--Hardy space, the variable Hardy space, the
Orlicz-slice Hardy space and their local versions. Some special cases of these
applications are even new and, particularly, in the case of the variable Hardy
space, the $g_\lambda^\ast$-function characterization obtained in this article
improves the known results via widening the range of $\lambda$.
