  Let $(X,d,\mu)$ be a space of homogeneous type, with the upper dimension
$\omega$, in the sense of R. R. Coifman and G. Weiss. Assume that $\eta$ is the
smoothness index of the wavelets on $X$ constructed by P. Auscher and T.
Hyt\"onen. In this article, when $p\in(\omega/(\omega+\eta),1]$, for the atomic
Hardy spaces $H_{\mathrm{cw}}^p(X)$ introduced by Coifman and Weiss, the
authors establish their various real-variable characterizations, respectively,
in terms of the grand maximal function, the radial maximal function, the
non-tangential maximal functions, the various Littlewood-Paley functions and
wavelet functions. This completely answers the question of R. R. Coifman and G.
Weiss by showing that no any additional (geometrical) condition is necessary to
guarantee the radial maximal function characterization of
$H_{\mathrm{cw}}^1(X)$ and even of $H_{\mathrm{cw}}^p(X)$ with $p$ as above. As
applications, the authors obtain the finite atomic characterizations of
$H^p_{\mathrm{cw}}(X)$, which further induce some criteria for the boundedness
of sublinear operators on $H^p_{\mathrm{cw}}(X)$. Compared with the known
results, the novelty of this article is that $\mu$ is not assumed to satisfy
the reverse doubling condition and $d$ is only a quasi-metric, moreover, the
range $p\in(\omega/(\omega+\eta),1]$ is natural and optimal.
