Live video streaming services have experienced significant growth since the emergence of social networking paradigms in recent years. In this scenario, adaptive bitrate streaming communications transmitted on web protocols provide a convenient and cost-efficient facility to serve various multimedia platforms over the Internet. In these communication models, video content is delivered optimally, possibly transcoded, edited automatically, and cached temporarily by network elements along the path. To this end, the computational capabilities of various network elements are considered as major resources to be optimized for service quality improvements. This article provides a contemporary survey of cutting-edge live video streaming studies from a computation-driven perspective. First, an overview of the global standards, system architectures, and streaming protocols is presented. Next, hierarchical computation-driven models of live video streaming are anatomized, including cloud-, edge-, and peer-to-peer-based solutions. Cutting-edge studies are then reviewed to discover the advances they have made in improving system performance in multiple areas. Finally, open challenges are presented to direct future research in this field.