  We consider the KdV equation $$ \partial_t u +\partial^3_x u +u\partial_x u=0
$$ with quasi-periodic initial data whose Fourier coefficients decay
exponentially and prove existence and uniqueness, in the class of functions
which have an expansion with exponentially decaying Fourier coefficients, of a
solution on a small interval of time, the length of which depends on the given
data and the frequency vector involved. For a Diophantine frequency vector and
for small quasi-periodic data (i.e., when the Fourier coefficients obey $|c(m)|
\le \varepsilon \exp(-\kappa_0 |m|)$ with $\varepsilon > 0$ sufficiently small,
depending on $\kappa_0 > 0$ and the frequency vector), we prove global
existence and uniqueness of the solution. The latter result relies on our
recent work \cite{DG} on the inverse spectral problem for the quasi-periodic
Schr\"{o}dinger equation.
