  In the first part of the paper we describe the complex geometry of the
universal Teichm\"uller space $\mathcal T$, which may be realized as an open
subset in the complex Banach space of holomorphic quadratic differentials in
the unit disc. The quotient $\mathcal S$ of the diffeomorphism group of the
circle modulo M\"obius transformations may be treated as a smooth part of
$\mathcal T$. In the second part we consider the quantization of universal
Teichm\"uller space $\mathcal T$. We explain first how to quantize the smooth
part $\mathcal S$ by embedding it into a Hilbert-Schmidt Siegel disc. This
quantization method, however, does not apply to the whole universal
Teichm\"uller space $\mathcal T$, for its quantization we use an approach, due
to Connes.
