The theory of dynamical systems has undergone a dramatical revolution in the 20th century. The beauty and power of the theory of dynamical systems is that it links together different areas of mathematics and physics. In the last 30 years a great deal of attention was dedicated to a statistical description of strange attractors. This led to the development of notions of various dimensions and entropies, which can be associated to the attractor, dynamical system or invariant measure. In this paper we review these notions and discuss relations between those, among which the most prominent is the so-called multifractal formalism.