We discuss problems concerning the conformal condenser capacity of “hedgehogs”, which are compact sets E in the unit disk D = {z: |z| < 1} consisting of a central body E₀ that is typically a smaller disk D_r = {z: |z| ≤ r}, 0 < r < 1, and several spikes E_k that are compact sets lying on radial intervals I(αk) = {te^iαk: 0 ≤ t < 1}. The main questions we are concerned with are the following: (1) How does the conformal capacity cap(E) of (E = ∪ⁿ_k=0E_k) behave when the spikes E_k, k = 1,…, n, move along the intervals I_(αk) toward the central body if their hyperbolic lengths are preserved during the motion? (2) How does the capacity cap(E) depend on the distribution of angles between the spikes E_k? We prove several results related to these questions and discuss methods of applying symmetrization type transformations to study the capacity of hedgehogs. Several open problems, including problems on the capacity of hedgehogs in the three-dimensional hyperbolic space, will also be suggested.