  Let $\mathfrak{D}$ be the space consists of pairs $(f,g)$, where $f$ is a
univalent function on the unit disc with $f(0)=0$, $g$ is a univalent function
on the exterior of the unit disc with $g(\infty)=\infty$ and
$f'(0)g'(\infty)=1$. In this article, we define the time variables $t_n, n\in
\Z$, on $\mathfrak{D}$ which are holomorphic with respect to the natural
complex structure on $\mathfrak{D}$ and can serve as local complex coordinates
for $\mathfrak{D}$. We show that the evolutions of the pair $(f,g)$ with
respect to these time coordinates are governed by the dispersionless Toda
hierarchy flows. An explicit tau function is constructed for the dispersionless
Toda hierarchy. By restricting $\mathfrak{D}$ to the subspace $\Sigma$ consists
of pairs where $f(w)=1/\bar{g(1/\bar{w})}$, we obtain the integrable hierarchy
of conformal mappings considered by Wiegmann and Zabrodin \cite{WZ}. Since
every $C^1$ homeomorphism $\gamma$ of the unit circle corresponds uniquely to
an element $(f,g)$ of $\mathfrak{D}$ under the conformal welding
$\gamma=g^{-1}\circ f$, the space $\text{Homeo}_{C}(S^1)$ can be naturally
identified as a subspace of $\mathfrak{D}$ characterized by $f(S^1)=g(S^1)$. We
show that we can naturally define complexified vector fields $\pa_n, n\in \Z$
on $\text{Homeo}_{C}(S^1)$ so that the evolutions of $(f,g)$ on
$\text{Homeo}_{C}(S^1)$ with respect to $\pa_n$ satisfy the dispersionless Toda
hierarchy. Finally, we show that there is a similar integrable structure for
the Riemann mappings $(f^{-1}, g^{-1})$. Moreover, in the latter case, the time
variables are Fourier coefficients of $\gamma$ and $1/\gamma^{-1}$.
