  Suppose that $(X,d,\mu)$ is a space of homogeneous type, with upper dimension
$\mu$, in the sense of R. R. Coifman and G. Weiss. Let $\eta$ be the H\"{o}lder
regularity index of wavelets constructed by P. Auscher and T. Hyt\"{o}nen. In
this article, the authors introduce the local Hardy space $h^{*,p}(X)$ via
local grand maximal functions and also characterize $h^{*,p}(X)$ via local
radial maximal functions, local non-tangential maximal functions, locally atoms
and local Littlewood--Paley functions. Furthermore, the authors establish the
relationship between the global and the local Hardy spaces. Finally, the
authors also obtain the finite atomic characterizations of $h^{*,p}(X)$. As an
application, the authors give the dual spaces of $h^{*,p}(X)$ when
$p\in(\omega/(\omega+\eta),1)$, which further completes the result of G. Dafni
and H. Yue on the dual space of $h^{*,1}(X)$. This article also answers the
question of R. R. Coifman and G. Weiss on the nonnecessity of any additional
geometric assumptions except the doubling condition for the radial maximal
function characterization of $H^1_{\mathrm{cw}}(X)$ when $\mu(X)<\infty$.
