  The theory of complex dimensions of fractal strings developed by Lapidus and
van Frankenhuijsen has proven to be a powerful tool for the study of Minkowski
measurability of fractal subsets of the real line. In a very general setting,
the Minkowski measurability of such sets is characterized by the structure of
corresponding complex dimensions. Also, this tool is particularly effective in
the setting of self-similar fractal subsets of $\mathbb{R}$ which have been
shown to be Minkowski measurable if and only if they are nonlattice. This paper
features a survey on the pertinent results of Lapidus and van Frankenhuijsen
and a preliminary extension of the theory of complex dimensions to subsets of
Euclidean space, with an emphasis on self-similar sets that satisfy various
separation conditions. This extension is developed in the context of
box-counting measurability, an analog of Minkowski measurability, which is
shown to be characterized by complex dimensions under certain mild conditions.
