In the last couple of years, academia-industry collaboration has demonstrated rapid advancements in the development of self-driving vehicles. Since it is evident that self-driving vehicles are going to reshape the traditional transportation systems in near future through enhancement in safe and smart mobility, motion control in self-driving vehicles while performing driving tasks in a dynamic road environment is still a challenging task. In this regard, we present a comprehensive study considering the evolution of steering control methods for self-driving vehicles. Initially, we discussed an insight into the traditional steering systems of the vehicles. To the best of our knowledge, currently, there is no taxonomy available, which elaborates steering control methods for self-driving vehicles. In this paper, we present a novel taxonomy including different steering control methods which are categorized into deterministic and heuristic steering control methods. Concurrently, the abovementioned techniques are critically reviewed elaborating their strengths and limitations. Based on the analysis, key challenges/research gaps in existing steering control methods along with the possible solutions have been briefly discussed to improve the effectiveness of the steering system of self-driving vehicles.