  We offer in this review a description of the vacuum energy of self-similar
systems. We describe two views of setting self-similar structures and point out
the main differences. A review of the authors' work on the subject is
presented, where they treat the self-similar system as a many-object problem
embedded in a regular smooth manifold. Focused on Dirichlet boundary
conditions, we report a systematic way of calculating the Casimir energy of
self-similar bodies where the knowledge of the quantum vacuum energy of the
single building block element is assumed and in fact already known. A
fundamental property that allows us to proceed with our method is the
dependence of the energy on a geometrical parameter that makes it possible to
establish the scaling property of self-similar systems. Several examples are
given. We also describe the situation, shown by other authors, where the
embedded space is a fractal space itself, having fractal dimension. A fractal
space does not hold properties that are rather common in regular spaces like
the tangent space. We refer to other authors who explain how some self-similar
configurations "do not have any smooth structures and one cannot define
differential operators on them directly". This gives rise to important
differences in the behavior of the vacuum.
