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.