Hyperelliptic Odd Coverings" are a class of odd coverings of C -> P^1, where C is a hyperelliptic curve. They are characterized by the behavior of the hyperelliptic involution of C with respect to an involution of P^1. I will talk about some ways for studying this type of coverings: by fixing an effective theta characteristic, they correspond to the solutions of a certain type of differential equations. Considering them as elements in a suitable Hurwitz space, they can be characterized using monodromy and then studied from the point of view of deformations. When C is general in H_g, the number of possible Hyperelliptic Odd Coverings C -> P^1 of minimum degree is finite. The main result will be how to compute this number. This is a work in collaboration with Gian Pietro Pirola. In the first part of the talk, I will introduce the Hyperelliptic Odd Coverings from a geometric point of view, contextualizing them in the panorama of other enumerative works (in collaboration with Farkas, Naranjo, Pirola, Lian). In the second part of the talk he will dedicate myself to deepening some demonstrations and some open problems subject to future analysis.