  We present a comprehensive investigation of $\epsilon$-entropy,
$h(\epsilon)$, in dynamical systems, stochastic processes and turbulence.
Particular emphasis is devoted on a recently proposed approach to the
calculation of the $\epsilon$-entropy based on the exit-time statistics. The
advantages of this method are demonstrated in examples of deterministic
diffusive maps, intermittent maps, stochastic self-affine and multi-affine
signals and experimental turbulent data. Concerning turbulence, the
multifractal formalism applied to the exit time statistics allows us to predict
that $h(\epsilon)\sim \epsilon^{-3}$ for velocity time measurement. This power
law is independent of the presence of intermittency and has been confirmed by
the experimental data analysis. Moreover, we show that the $\epsilon$-entropy
density of a 3-dimensional velocity field is affected by the correlations
induced by the sweeping of large scales.
