<p>This expository paper is written in remembrance of Igor Borisovich Simonenko (1935–2008) on the 90th anniversary of his birth. In the 1960s, Simonenko made fundamental contributions to one- and higher-dimensional convolution operators. The focus of this paper is on quarter-plane Toeplitz operators, the most prominent operators among multidimensional discrete convolutions. I will survey in a gentle way several selected topics around Simonenko’s Fredholm criterion of 1968, including the role played by local sectoriality and the invertibility problem for Toeplitz operators on the quarter-plane.</p>