Let $\mu$ be a probability measure on a metric space $(E,d)$ and $N$ a positive integer.
The {\em quantization error} $e_N$ of $\mu$ is defined as the infimum over all subsets ${\cal{E}}$
of $E$ of cardinality $N$ of the average distance w.~r.~t.~$\mu$ to the closest point in the set
${\cal{E}}$. We study the asymptotics of $e_N$ for large $N$. We concentrate on the
case of a Gaussian measure $\mu$ on a Banach space. The asymptotics of $e_N$ is closely related to
{\em small ball probabilities} which have received considerable interest in the past decade.
The quantization problem is motivated by the problem of encoding a continuous signal
by a specified number of bits with minimal distortion. This is joint work with Steffen Dereich,
Franz Fehringer, Anis Matoussi and Michail Lifschitz.