Abstract The atom-bond sum-connectivity (ABS) index of a graph G G with edges e 1 , … , e m {e}_{1},\ldots ,{e}_{m} is the sum of the numbers 1 − 2 ( d e i + 2 ) − 1 \sqrt{1-2{\left({d}_{{e}_{i}}+2)}^{-1}} over 1 ≤ i ≤ m 1\le i\le m , where d e i {d}_{{e}_{i}} is the number of edges adjacent to e i {e}_{i} . In this article, we study the maximum values of the ABS index over graphs with given parameters. More specifically, we determine the maximum ABS index of connected graphs of a given order with a fixed (i) minimum degree, (ii) maximum degree, (iii) chromatic number, (iv) independence number, or (v) number of pendent vertices. We also characterize the graphs attaining the maximum ABS values in all of these classes.