  Our main goal in this long survey article is to provide an overview of the
theory of complex fractal dimensions and of the associated geometric or fractal
zeta functions, first in the case of fractal strings (one-dimensional drums
with fractal boundary), in \S2, and then in the higher-dimensional case of
relative fractal drums and, in particular, of arbitrary bounded subsets of
Euclidean space of $\mbr^N$, for any integer $N \geq 1$, in \S3.
  Special attention is paid to discussing a variety of examples illustrating
the general theory rather than to providing complete statements of the results
and their proofs, for which we refer to the author's previous (joint) books
mentioned in the paper.
  Finally, in an epilogue (\S4), entitled "From quantized number theory to
fractal cohomology", we briefly survey aspects of related work (motivated in
part by the theory of complex fractal dimensions) of the author with H. Herichi
(in the real case) [HerLap1], along with [Lap8], and with T. Cobler (in the
complex case) [CobLap1], respectively, as well as in the latter part of a book
in preparation by the author, [Lap10].
