The atom-bond connectivity index (ABC-index) is a useful topological index employed in studying the stability of alkanes and the strain energy of cycloalkanes. The ABC-index of a nontrivial graph G, denoted by ABC(G), is defined as ABC(G)=∑vivj∈E(G)di+dj−2didj, where di is the degree of vertex vi in G. In this paper, we first present some lower bounds for ABC-index by means of some known inequalities. Then we give some upper bounds for ABC-index in terms of graph parameters such as clique number, vertex connectivity, algebraic connectivity and spectral radius. Moreover, we give some lower and upper bounds for ABC-index of Mycielski graphs. Finally, we compare ABC-index with other graph invariants for connected graphs.