Conversational Machine Comprehension (CMC) is a research track in conversational AI which expects the machine to understand an open-domain text and thereafter engage in a multi-turn conversation to answer questions related to the text. While most of the research in Machine Reading Comprehension (MRC) revolves around single-turn question answering, multi-turn CMC has recently gained prominence, thanks to the advancement in natural language understanding via neural language models like BERT and the introduction of large-scale conversational datasets like CoQA and QuAC. The rise in interest has, however, led to a flurry of concurrent publications, each with a different yet structurally similar modeling approach and an inconsistent view of the surrounding literature. With the volume of model submissions to conversational datasets increasing every year, there exists a need to consolidate the scattered knowledge in this domain to streamline future research. This literature review, therefore, is a first-of-its-kind attempt at providing a holistic overview of CMC, with an emphasis on the common trends across recently published models, specifically in their approach to tackling conversational history. It focuses on synthesizing a generic framework for CMC models, rather than describing the models individually. The review is intended to serve as a compendium for future researchers in this domain.