In this paper we extend Shishikura's result on the Hausdorff dimension of the boundary of the Mandelbrot set to higher-dimensional parameter spaces, for example the space of degree d polynomials, and show that some parameter subsets, including the boundary of the connectedness locus, have Hausdorff dimensions equal to the real dimension of the parameter space (which is four for cubic polynomials). PACS number: 0545