In the vehicle routing problem with simultaneous pickup and delivery (VRPSPD), goods have to be transported from different origins to different destinations, and each customer has both a delivery and a pickup demand to be satisfied simultaneously. The VRPSPD has been around for about 30 years, and significant progress has since been made on this problem and its variants. This paper aims to comprehensively review the existing work on the VRPSPD. It surveys mathematical formulations, algorithms, variants, case studies, and industrial applications. It also provides an overview of trends in the literature and identifies several interesting promising future research perspectives.