HTTP Adaptive Streaming (HAS) is the most recent attempt regarding video quality adaptation. It enables cheap and easy to implement streaming technology without the need for a dedicated infrastructure. By using a combination of TCP and HTTP it has the advantage of reusing all the existing technologies designed for ordinary web usage. Equally important is that HAS traffic passes through firewalls and works well when NAT is deployed. The rate adaptation controller of HAS, commonly called Adaptive Bitrate Selection (ABR), is currently receiving a lot of attention from both industry and academia. However, most of the research efforts concentrate on a specific aspect or a particular methodology without considering the overall context. This paper presents a comprehensive survey of the most significant research activities in the area of client-side HTTP-based adaptive video streaming. It starts by decomposing the ABR module into three subcomponents, namely: resource estimation function, chunk request scheduling, and adaptation module. Each subcomponent encapsulates a particular function that is vital for the operation of an ABR scheme. A review of each of the subcomponents and how they interact with each other is presented. Furthermore, those external factors that are known to have a direct impact on the performance of an ABR module, like content nature, CDN, and context are discussed. In conclusion, the paper provides an extensive reference for further research in the field.