  Let $D$ be a domain in the complex plane $\mathbb{C}$. The Hardy number of
$D$, which first introduced by Hansen, is the maximal number $h(D)$ in
$[0,+\infty]$ such that $f$ belongs to the classical Hardy space $H^p
(\mathbb{D})$ whenever $0<p<h(D)$ and $f$ is holomorphic on the unit disk
$\mathbb{D}$ with values in $D$. As an analogue notion to the Hardy number of a
domain $D$ in $\mathbb{C}$, we introduce the Bergman number of $D$ and we
denote it by $b(D)$. Our main result is that, if $D$ is regular, then
$h(D)=b(D)$. This generalizes earlier work by the author and Karamanlis for
simply connected domains. The Bergman number $b(D)$ is the maximal number in
$[0,+\infty]$ such that $f$ belongs to the weighted Bergman space $A^p_{\alpha}
(\mathbb{D})$ whenever $p>0$ and $\alpha>-1$ satisfy
$0<\frac{p}{\alpha+2}<b(D)$ and $f$ is holomorphic on $\mathbb{D}$ with values
in $D$. We also establish several results about Hardy spaces and weighted
Bergman spaces and we give a new characterization of the Hardy number and thus
of the Bergman number of a regular domain with respect to the harmonic measure.
