The fifth generation (5G) cellular network aims at providing very high data rates, ultra reliable low latency communications, and a vast increase of connection density. As one of the design trends towards these objectives, 5G exploits multi-connectivity, i.e., the concurrent use of multiple access networks. The Access Traffic Steering, Switching, and Splitting (ATSSS) architecture has recently been proposed to enable 5G multi-connectivity, and multipath transport protocols have emerged as a key ATSSS technology enabler. Within this context, this survey presents a detailed review of multipath transport protocols, identifies their existing and potential exploitation in ATSSS, and suggests their applicability for enhanced Mobile Broadband (eMBB) and Ultra Reliable Low Latency Communications (URLLC) services. To this end, we first review 5G background and current standardization activities around multiconnectivity and the ATSSS architecture. We then provide an in-depth review of multipath transport protocols, covering four core functionalities, i.e., path management, scheduling, congestion control, and reliable transfer. Based on the reviewed literature, we further discuss the integration of multipath transport into ATSSS to achieve eMBB and URLLC service requirements. Finally, we also point out major open research issues and discuss possible future directions.INDEX TERMS Multipath transport protocols, ATSSS (Access Traffic Steering, Switching and Splitting), eMBB (enhanced Mobile Broadband), URLLC (Ultra Reliable Low Latency Communication).