We present the background on classical real hedgehogs, from their genesis as sums and formal differences of convex bodies. We see that hedgehogs may be defined in several different ways: as envelopes parametrized by their Gauss map, if the support function is \(\mathrm {C}^2\) ; as Legendrian fronts, if the support function is \(\mathrm {C}^{\infty }\) ; via Euler Calculus, if the support function is analytic; and, most generally, when the support function is a difference of support functions of arbitrary convex bodies, inductively with respect to the dimension, replacing support sets by “support hedgehogs”. Examples, properties and tools are given for each variant. Different types of indexes of a point with respect to a hedgehog are introduced, with applications. We say a few words on projective and polarity dualities applied to hedgehogs. The presentation of \(\mathrm {C}^{\infty }\) -hedgehogs as Legendrian fronts is used to describe their generic singularities, leading to a first open problem raised by R. Langevin, G. Levitt and H. Rosenberg: Does there exist a projective hedgehog without any swallowtail in \(\mathbb {R}^{3}\) ?