Topological indices such as the general Sombor index are studied because of their extensive applications. For c, g ∈ ℝ, the general Sombor index for a graph H is SOc,g(H) = ∑vw∈E(H) ([dH(v)]c +[dH(w)c])g, where E(H) is the set of edges of H, and dH(v) and dH(w) are the degrees of vertices v and w. Trees of given order with the largest SOc,g for c ≥ 1 and g > 0, trees of given order with the smallest SOc,g for c ≥ 1 and g ≥ 1, bipartite graphs of prescribed matching number and order with the largest SOc,g for c ≥ 1 and g ≥ 0 are presented. We also obtain several corollaries including bounds for the classical Sombor index and forgotten index.