In this talk, we will define and compare different definitions
of dimension (Hausdorff, Minkowski, and packing) used to analyze fractal
sets. Then we will give a survey of several results about these
dimensions in the context of Julia sets of entire functions. The type of
entire function we are studying makes a tremendous difference; the
results are quite different if the Julia set is associated with a
polynomial versus a Julia set associated with a transcendental
(non-polynomial) entire function. We will conclude by discussing my
recent construction of Julia sets of transcendental entire functions
with packing dimension strictly between one and two.