A huge amount of generated data is regularly exploding into the network by the users through smartphones, laptops, tablets, self-configured Internet-of-things (IoT) devices, and machine-to-machine (M2M) communication. In such a situation, satisfying critical quality-of-service (QoS) requirements (e.g., throughput, latency, bandwidth, and reliability) is a large challenge as a vast amount of data travels into the network. Nowadays, strict QoS requirements must be satisfied efficiently in many networked multimedia applications when intelligent multi-homed devices are used. Such devices support the concept of multi-homing. To be precise, they have multiple network interfaces that aim to connect and communicate concurrently with different networking technologies. Therefore, many multipath transport protocols are provided to multi-homed devices, which aim (1) to take advantage of several network paths at the transport layer (Layer-4) and (2) to meet the strict QoS requirements for providing low network latency, higher data rates, and increased reliability. To this end, this survey first presents the challenges/problems for supporting multipath transmission with possible solutions. Then, it reviews recent research efforts related to the concurrent multipath transmission (CMT) protocol and the multipath transmission control protocol (MPTCP). It reviews the latest research efforts by considering (1) how a multipath transport protocol operates (i.e., its functionality); (2) in what type of network; (3) what path characteristics it should consider; and (4) how it addresses various design challenges. Furthermore, it presents some lessons learned and discusses open research issues in multipath transport protocols.