Let G = (V, E) be a (molecular) graph. For a family of graphs G, the first Zagreb index M1 and the second Zagreb index M2 have already studied. In particular, it has been presented, the first Zagreb index M1 and the second Zagreb index M2 of trees T in terms of domination parameter. In this paper, we present upper bounds on Zagreb indices of unicyclic and bicyclic graphs with a given domination number and also find upper bounds on the Zagreb indices of trees, unicyclic, and bicyclic graphs with a given total domination number.