Four-dimensional black holes (BH) of GR do not deform under external fields. This is a very special result of 4D, and ceases to hold in higher dimensions. In this talk I will discuss the static gravitational and magnetic response of charged BHs in arbitrary spacetime dimensions D. First, I will derive exact analytical expressions for the tensor Love numbers in terms of the BH temperature. These turn out to vanish at extremality. In contrast, vector Love numbers and magnetic susceptibilities increase significantly as one approaches the maximally charged limit. I will provide exact analytical expressions at extremality and solve numerically the finite-temperature and finite-charge case. In D=4 all response parameters remain zero at all temperatures. I will conclude by discussing several aspects that could be explored in future work.