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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.
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