We give a short introduction to hedgehog theory in non-Euclidean spaces. Drawing on work of F. Fillastre, we give a detailed study of plane Lorentzian and Fuchsian hedgehogs, including a series of Fuchsian analogues of classical geometrical inequalities (which are reversed, compared to classical ones). We define hedgehogs in real projective space \(\mathbb {R}P^{n+1}\) and hyperbolic space \(\mathbb {H}^{n+1}\) and distinguish between two types of hedgehogs in \(\mathbb {H}^{n+1}: \mathrm {g} \) -hedgehogs, which are envelopes of smooth families of cooriented (totally geodesic) hyperplanes of \(\mathbb {H}^{n+1}\) , where g indicates “geodesically hedgehog hypersurfaces”; and h-hedgehogs, where horospheres play in \(\mathbb {H}^{n+1}\) the role assigned to cooriented hyperplanes in \(\mathbb {R}^{n+1}\) , and the corresponding ideal points the role of the unit normal vectors. These two notions of hedgehogs correspond to the two natural notions of convexity in hyperbolic space: geodesical convexity and the stronger horospherical convexity. We define the signed h-width of an h-hedgehog in the direction of an arbitrary ideal point in the ideal boundary sphere at infinity of \(\mathbb {H}^{n+1}\) and then give a simple condition for an h-hedgehog of \(\mathbb {H}^{n+1}\) to be of constant h-width.