  In this article, the authors establish the wavelet characterization of Besov
and Triebel--Lizorkin spaces on a given space $(X,d,\mu)$ of homogeneous type
in the sense of Coifman and Weiss. Moreover, the authors introduce almost
diagonal operators on Besov and Triebel--Lizorkin sequence spaces on $X$, and
obtain their boundedness. Using this wavelet characterization and this
boundedness of almost diagonal operators, the authors obtain the molecular
characterization of Besov and Triebel--Lizorkin spaces. Applying this molecular
characterization, the authors further establish the Littlewood--Paley
characterizations of Triebel--Lizorkin spaces on $X$. The main novelty of this
article is that all these results get rid of their dependence on the reverse
doubling property of $\mu$ and also the triangle inequality of $d$, by fully
using the geometrical property of $X$ expressed via its equipped quasi-metric
$d$, dyadic reference points, dyadic cubes, and wavelets
