Multiferroics are a time-honoured research subject by reason for their tremendous application potential in the information industry, such as in multi-state information storage devices and new types of sensors. An outburst of studies on multiferroicity has been witnessed in the 21st century, although this field has a long research history since the 19th century. Multiferroicity has now become one of the hottest research topics in condensed matter physics and materials science. Numerous efforts have been made to investigate the cross-coupling phenomena among ferroic orders such as ferroelectricity, (anti-)ferromagnetism, and ferroelasticity, especially the coupling between electric and magnetic orderings that would account for the magnetoelectric (ME) effect in multiferroic materials. The magnetoelectric properties and coupling behavior of single phase multiferroics are dominated by their domain structures. It was also noted that, however, the multiferroic materials exhibit very complicated domain structures. Studies on domain structure characterization and domain switching are a crucial step in the exploration of approaches to the control and manipulation of magnetic (electric) properties using an electric (magnetic) field or other means. In this review, following a concise outline of our current basic knowledge on the magnetoelectric (ME) effect, we summarize some important research activities on domain switching in single-phase multiferroic materials in the form of single crystals and thin films, especially domain switching behavior involving strain and the related physics in the last decade. We also introduce recent developments in characterization techniques for domain structures of ferroelectric or multiferroic materials, which have significantly advanced our understanding of domain switching dynamics and interactions. The effects of a series of issues such as electric field, magnetic field, and stress effects on domain switching are been discussed as well. It is intended that an integrated viewpoint of these issues, as provided here, will further motivate synergistic activities between the various research groups and industry towards the development and characterization of multiferroic materials.