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Our main goal in this long survey article is to provide an overview of thetheory of complex fractal dimensions and of the associated geometric or fractalzeta functions, first in the case of fractal strings (one-dimensional drumswith fractal boundary), in \S2, and then in the higher-dimensional case ofrelative fractal drums and, in particular, of arbitrary bounded subsets ofEuclidean space of $\mbr^N$, for any integer $N \geq 1$, in \S3.  Special attention is paid to discussing a variety of examples illustratingthe general theory rather than to providing complete statements of the resultsand their proofs, for which we refer to the author's previous (joint) booksmentioned in the paper.  Finally, in an epilogue (\S4), entitled "From quantized number theory tofractal cohomology", we briefly survey aspects of related work (motivated inpart 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 thecomplex case) [CobLap1], respectively, as well as in the latter part of a bookin preparation by the author, [Lap10].

An Overview of Complex Fractal Dimensions: From Fractal Strings to Fractal Drums, and Back