This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.
The Mandelbrot Set has a dimension of 2. The Mandelbrot Set
contains and is contained in a disk. A disk has a dimension of 2, thus
so does the Mandelbrot Set.
The Koch snowflake (Hausdorff dimension 1.2619...) does not satisfy
this condition because it is a thin boundary curve, thus containing no
disk. If you add the region inside the curve then it does have
dimension of 2.
The boundary of the Mandelbrot set and the Julia set of a generic c in
M have Hausdorff dimension 2 and have topological dimension 1. The
proof is based on the study of the bifurcation of parabolic periodic
points. (Since the boundary has empty interior, the topological
dimension is less than 2, and thus is 1.) See reference above