  A Hardy inequality of the form \[\int_{\tilde{\Omega}} |\nabla f({\bf{x}})|^p
d {\bf{x}} \ge (\frac{p-1}{p})^p \int_{\tilde{\Omega}} \{1 + a(\delta, \partial
\tilde{\Omega})(\x)\}\frac{|f({\bf{x}})|^p}{\delta({\bf{x}})^p} d{\bf{x}}, \]
for all $f \in C_0^{\infty}({\tilde{\Omega}})$, is considered for $p\in
(1,\infty)$, where ${\tilde{\Omega}}$ can be either $\Omega$ or $\mathbb{R}^n
\setminus \Omega$ with $\Omega$ a domain in $\mathbb{R}^n$, $n \ge 2$, and
$\delta({\bf{x}})$ is the distance from ${\bf{x}} \in {\tilde{\Omega}} $ to the
boundary $ \partial {\tilde{\Omega}}.$ The main emphasis is on determining the
dependance of $a(\delta, \partial {\tilde{\Omega}})$ on the geometric
properties of $\partial {\tilde{\Omega}}.$ A Hardy inequality is also
established for any doubly connected domain $\Omega$ in $\mathbb{R}^2$ in terms
of a uniformisation of $\Omega,$ that is, any conformal univalent map of
$\Omega$ onto an annulus.
