We study the density of polynomials in
H
2
(
Ω
,
e
−
φ
)
H^2(\Omega ,e^{-\varphi })
, the space of square integrable holomorphic functions in a bounded domain
Ω
\Omega
in
C
\mathbb {C}
, where
φ
\varphi
is a subharmonic function. In particular, we prove that the density holds in Carathéodory domains for any subharmonic function
φ
\varphi
in a neighborhood of
Ω
¯
\overline {\Omega }
. In non-Carathéodory domains, we prove that the density depends on the weight function, giving examples.