  One defines a non-homogeneous space $(X, \mu)$ as a metric space equipped
with a non-doubling measure $\mu$ so that the volume of the ball with center
$x$, radius $r$ has an upper bound of the form $r^n$ for some $n> 0$. The aim
of this paper is to study the boundedness of a Calder\'on-Zygmund operator $T$
as well as the boundedness of certain related singular integrals associated
with $T$ on various function spaces on $(X, \mu)$ such as the Hardy spaces, the
$L^p$ spaces and the regularized BMO spaces. This article thus extends the work
of X. Tolsa \cite{T1} on the non-homogeneous space $(\mathbb R^n, \mu)$ to the
setting of a general non-homogeneous space $(X, \mu)$. While our framework is
similar to that of \cite{H}, we are able to obtain quite a few properties
similar to those of Calder\'on-Zygmund operators on doubling spaces, including
the following for such an operator $T$: weak type $(1,1)$ estimate, boundedness
from Hardy space into $L^1$, boundedness from $L^{\infty}$ into the regularized
BMO and an interpolation theorem. We also prove that the dual space of the
Hardy space is the regularized BMO space, obtain a Calder\'on-Zygmund
decomposition on the non-homogeneous space $(X, \mu)$ and use this
decomposition to show the boundedness of the maximal operators in the form of
Cotlar inequality as well as the boundedness of commutators of
Calder\'on-Zygmund operators and BMO functions.
