In recent years, there has been a growing interest in designing multi-robot systems (hereafter MRSs) to provide cost effective, fault-tolerant and reliable solutions to a variety of automated applications. Here, we review recent advancements in MRSs specifically designed for cooperative object transport, which requires the members of MRSs to coordinate their actions to transport objects from a starting position to a final destination. To achieve cooperative object transport, a wide range of transport, coordination and control strategies have been proposed. Our goal is to provide a comprehensive summary for this relatively heterogeneous and fast-growing body of scientific literature. While distilling the information, we purposefully avoid using hierarchical dichotomies, which have been traditionally used in the field of MRSs. Instead, we employ a coarse-grain approach by classifying each study based on the transport strategy used; pushing-only, grasping and caging. We identify key design constraints that may be shared among these studies despite considerable differences in their design methods. In the end, we discuss several open challenges and possible directions for future work to improve the performance of the current MRSs. Overall, we hope to increase the visibility and accessibility of the excellent studies in the field and provide a framework that helps the reader to navigate through them more effectively.