Perfect matchings in chemical graphs correspond to Kekulé structures in underlying chemical structure, which play an important role in the analysis of resonance energy and stability of certain chemical compounds. In 2007, Vukičević and Trinajstić introduced the concept of the anti-Kekulé number to be the smallest number of edges that have to be removed from a graph such that it remains connected but without any perfect matching. Thereafter, this parameter attracted much more attention of researchers both in mathematics and theoretical chemistry. In the present paper, we report on the results related to the anti-Kekulé of graphs. Some new results on the anti-Kekulé number of nanotubes, polyomino system and cactus chains are also proven. We conclude the paper with some open problems.