  Let $({\mathcal X}, d, \mu)$ be a metric measure space and satisfy the
so-called upper doubling condition and the geometrical doubling condition. In
this paper, we introduce the atomic Hardy space $H^1(\mu)$ and prove that its
dual space is the known space ${\rm RBMO}(\mu)$ in this context. Using this
duality, we establish a criterion for the boundedness of linear operators from
$H^1(\mu)$ to any Banach space. As an application of this criterion, we obtain
the boundedness of Calder\'on--Zygmund operators from $H^1(\mu)$ to $L^1(\mu)$.
