Surface nanopatterning induced by ion beam irradiation (IBI) has emerged as an effective nanostructuring technique since it induces patterns on large areas of a wide variety of materials, in short time, and at low cost. Nowadays, two main subfields can be distinguished within IBI nanopatterning depending on the irrelevant or relevant role played by the surface composition. In this review, we give an up-dated account of the progress reached when surface composition plays a relevant role, with a main focus on IBI surface patterning with simultaneous co-deposition of foreign atoms. In addition, we also review the advances in IBI of compound surfaces as well as IBI systems where the ion employed is not a noble gas species. In particular, for the IBI with concurrent metal co-deposition, we detail the chronological evolution of these studies because it helps us to clarify some contradictory early reports. We describe the main patterns obtained with this technique as a function of the foreign atom deposition pathway, also focusing in those systematic studies that have contributed to identify the main mechanisms leading to the surface pattern formation and development. Likewise, we explain the main theoretical models aimed at describing these nanopattern formation processes. Finally, we address two main special features of the patterns induced by this technique, namely, the enhanced pattern ordering and the possibility to produce both morphological and chemical patterns.