This work studies the scattering problem on the real axis for the Sturm-Liouville equation with discontinuous leading coefficient and the real-valued steplike potential q(x) that has different constant asymptotes as x +/-. We investigate the properties of the scattering data, obtain the main integral equations of the inverse scattering problem, and also give necessary and sufficient conditions characterizing the scattering data. Copyright (c) 2013 John Wiley & Sons, Ltd.