  Let $({\mathcal X}, d, \mu)$ be a separable metric measure space satisfying
the known upper doubling condition, the geometrical doubling condition and the
non-atomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$. In this
paper, we show that the boundedness of a Calder\'on-Zygmund operator $T$ on
$L^2(\mu)$ is equivalent to that of $T$ on $L^p(\mu)$ for some $p\in (1,
\infty)$, and that of $T$ from $L^1(\mu)$ to $L^{1,\,\infty}(\mu).$ As an
application, we prove that if $T$ is a Calder\'on-Zygmund operator bounded on
$L^2(\mu)$, then its maximal operator is bounded on $L^p(\mu)$ for all $p\in
(1, \infty)$ and from the space of all complex-valued Borel measures on
${\mathcal X}$ to $L^{1,\,\infty}(\mu)$. All these results generalize the
corresponding results of Nazarov et al. on metric spaces with measures
satisfying the so-called polynomial growth condition.
