In recent decades, there are immense applications for bulk and few-layer magnetic insulators in biomedicine, data storage, and signal transfer. In these applications, the interaction between spin and lattice vibration has significant impacts on the device performance. In this article, we systematically review the fundamental physical aspects of magnon-phonon coupling in magnetic insulators. We first introduce the fundamental physics of magnons and magnon-phonon coupling in magnetic insulators and then discuss the influence of magnon-phonon coupling on the properties of magnons and phonons. Finally, a summary is presented, and we also discuss the possible open problems in this field. This article presents the advanced understanding of magnon-phonon coupling in magnetic insulators, which provides new opportunities for improving various possible applications.