Throughout this survey, let G denote a non-elementary, geometrically finite Kleinian group. It is well known that the discontinuous action of such a group on hyperbolic space gives rise to the perfect limit set L(G) of G. The set L(G) has rather interesting fractal properties, and the purpose of these notes will be to recall and clarify a few of these. In particular, we shall specify various fractal dimensions of the limit set, such as Hausdorff dimension, packing dimension and box-counting dimension, as well as the generalised q-th Renyi dimension, the q-th logarithmic index and the information dimension, with respect to the associated Patterson measure and for non-negative q. The derivation of these fractal quantities relies on a comprehensive understanding of the Patterson measure (specifically, the 'global measure formula' and the 'Khintchine-like law of iterated logarithm'), and as a corollary we obtain the result that this particular conformal measure is a regular measure, but in fact not a fractal measure. Furthermore, in the 2- and 3-dimensional case we relate the Patterson measure to the geometric measures Hausdorff measure and packing measure. Although the Patterson measure is not fractal, it still makes sense to consider certain weak singularity spectra of this measure. By using results from the metrical Diophantine analysis of Kleinian limit sets, we derive these spectra for the Patterson measure. For simplicity, we restrict the latter investigations to the special case in which G is a finitely generated 2nd kind Fuchsian group with parabolic elements. We remark that analogous multiple fractal statements may be derived for geometrically finite Julia sets of rational functions. This follows since for such Julia sets there exist conformal measures with essentially the same properties as in the Kleinian case.