High-quality services over wired and wireless networks require high bitrate network content delivery. Multipath transport protocols utilize multiple paths to support increased throughput for network applications. Ubiquitous mobile devices with multiple interfaces such as WiFi and 4G/5G cellular, and data centers supporting big data analysis and cloud-computing, motivate adoption of multipath transmission in current and future network architectures. Congestion control mechanisms play a fundamental role in these multipath transport protocols. Diverse approaches have been proposed in the literature, differing in terms of their goals, principles, performance, and mostly in how various issues are addressed in their design. This paper presents a survey of congestion control solutions for multipath transport protocols and discusses the multipath congestion control design in order to address the need for some desirable properties including TCP-friendliness, load balancing, stability, and Pareto optimality. Existing window-based and rate-based multipath congestion control algorithms are investigated and categorized based on their theoretical algorithm design. Next, this paper discusses the congestion control mechanisms used in diverse multipath transport protocols in the context of different network scenarios. Finally, the trends in the future multipath congestion control design are presented.