Machine Translation (MT), the way of translating texts or documents from a source language to a target language automatically without human intervention, has gained popularity in the growing information technology-based era of globalization. Bangla is a major language, and several MT studies with different tools and techniques have been investigated in the last two decades. Considering the importance of the Bangla language and its prospects in MT studies, this study provides a comprehensive review of existing Bangla MT studies to meet the timely demand. Specifically, at first, the basic ideas of different MT methods (Rule-based, Example-based, Statistical, Neural, and Hybrid) and performance measures of MT are presented as a background study of the present review. Then an overview of the Bangla language and a brief description of the available Bangla-English corpora are provided. Next, a description of the existing Bangla MT studies is provided categorically following the common strategic fashion to create a valuable reference for current researchers in the field that is also suitable for nonexpert users. The achieved performances of individual methods are also compared in a tabular form. Finally, a number of future research prospects are revealed from the studies, encouraging researchers and practitioners to develop a better and comprehensive Bangla MT system.