We have continued to employ the concepts of Mandelbrot's fractal geometry to the quantitative studies of central nervous system neurons, and other cell types grown in tissue culture or from whole animals. We do this by employing image processing techniques to measure the fractal dimension (D), which is a measure of the complexity of the structure under investigation. In particular, the D relates to the degree of branching (e.g., of dendrites), the ruggedness of borders and the degree of space-filling of the object of interest. We have recently begun a study of the effects of growth substrate on the differentiation of cultured neurons and will use the fractal dimension to quantitate any differences found. For some time it has been recognized that there is a general problem in fractal geometry with the global nature of D, in that it says little about the local details of the structure of an object. thus, two fractal objects may have very different structures but have the same D. We have developed methods that measure the mass dimension and lacunarity that provide distinguishing measures in such cases. We have also developed methods for measuring the generalized dimensions, the D(q)s, which are other distinguishing measures.