We give some characterizations of ϒ -hyperelliptic Riemann surfaces of genus ɡ ≥ 2, that is, pairs (S, j) where S is a closed Riemann surface of genus ɡ and j : S ⟶ S is a conformal involution with exactly 2ɡ + 2 - 4ϒ  fixed points. These characterizations are given by Schottky groups, special hyperbolic polygons and algebraic curves.These can be seen as generalizations of the works [5] and [11].