Wireless technology, especially Vehicular Ad-hoc Network (VANET), is developing rapidly. VANET is an emerging technology that assists intelligent transportation systems by improving traffic services and helping in minimizing road accidents. Data sharing in VANETs is time-critical, necessitating the formation of fast and robust network connections. Due to the highly dynamic nature of VANET, providing reliable, consistent, and seamless communication is a significant challenge. In the last decade, various routing approaches have been proposed to efficiently handle quick handover of safety and infotainment-related VANET applications. This paper reviews and investigates the existing routing protocols and classifies them into a taxonomy based on essential attributes such as forwarding strategies, routing strategies, network dimensions. Routing challenges and future research directions in the VANET area are discussed in this paper.