Multipathing in communication networks is gaining momentum due to its attractive features of increased reliability, throughput, fault tolerance, and load balancing capabilities. In particular, wireless environments and datacenters are envisioned to become largely dependent on the power of multipathing for seamless handovers, virtual machine (VM) migration and in general, pooling less proficient resources together for achieving overall high proficiency. The transport layer, with its knowledge about end-to-end path characteristics, is well placed to enhance performance through better utilization of multiple paths. Realizing the importance of transport-layer multipath, this paper investigates the modernization of traditional connection establishment, flow control, sequence number splitting, acknowledgement, and flow scheduling mechanisms for use with multiple paths. Since congestion control defines a fundamental feature of the transport layer, we study the working of multipath rate control and analyze its stability and convergence. We also discuss how various multipath congestion control algorithms differ in their window increase and decrease functions, their TCP-friendliness, and responsiveness. To the best of our knowledge, this is the first in-depth survey paper that has chronicled the evolution of the transport layer of the Internet from the traditional single-path TCP to the recent development of the modern multipath TCP (MPTCP) protocol. Along with describing the history of this evolution, we also highlight in this paper the remaining challenges and research issues.