The constant desire for faster data rates, lower latency, improved reliability, global device integration, and pervasiveness are some of the factors driving the development of next-generation communication systems. Sixth-generation (6G) networks have received a lot of attention from the industry and academics as fifth-generation (5G) communications are being rolled out globally. With the proliferation of smart devices and the Internet of Things (IoT), 6G networks will require ultra-reliable and low-latency communication. Routing protocols have a significant role in improving the performance of a network. Traditional routing techniques will have difficulty coping with the highly complex and dynamic 6G environments. Recently, machine learning (ML), a key component of artificial intelligence, is emerging as the key to managing complex and dynamic networks efficiently. However, there are still several significant challenges that need to be addressed. In this paper, we provide an overview of current machine-learning techniques used in network routing. Lastly, we highlight open research problems that need to be addressed and prospects for future research.