Instability of orbits in dynamical systems causes complexity of their
behavior. It can be measured by different characteristics. The
epsilon-complexity is one of them, and it is very important because of the
following: 1) it is a pre-limiting quantity--it contains an information
not only about asymptotics but about fluctuations as well; 2) there is a direct link to fractal dimensions and topological entropy; 3) it works
not only in hyperbolic situations but also in polynomial ones. In the talk, it is supposed to discuss main properties of the
epsilon-complexity and some recent results in this area.
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