In this paper, we extend the recently introduced vertex-degree-based topological index, the Sombor index, and we call it general Sombor index. The general Sombor index generalizes both the forgotten index and the Sombor index. We present the bounds in terms of other important graph parameters for general Sombor index. We also explore the Nordhaus–Gaddum-type result for the general Sombor index. We present further the relations between general Sombor index and other generalized indices: general Randić index and general sum-connectivity index.