With the deployment of multihoming and multipath routing, multipath between wired Internet hosts becomes more and more common. Accompanied by the rapid development of various wireless transmission technologies in recent years, wireless access has become a main access method of the Internet. The swift increase on wireless terminals with multiple network interfaces and the prompt spread of various heterogeneous wireless networks have made it possible for terminals to carry out concurrent multipath transfer. Concurrent multipath transfer can achieve the bandwidth aggregation of multipath and thus greatly improve network utilization, and therefore has become a hot research topic. Meanwhile, the diversity of multipath poses challenges to multipath transport protocol design, such as how to deal with the receiving buffer blockings caused by out-of-order delivery of data packets, the throughput suppression caused by improper multipath packet scheduling, the bandwidth underutilization caused by improper cooperation of multipath, and unfair resource allocation caused by multipath heterogeneity. In this paper, how the existing reliable multipath transport protocols solve the above challenges is surveyed. In conclusion, none of these existing protocols have completely solved the above challenges. Some open research problems and future trend are also stated in the paper.