Machine translation (MT) plays an important role in benefiting linguists, sociologists, computer scientists, etc. by processing natural language to translate it into some other natural language. And this demand has grown exponentially over past couple of years, considering the enormous exchange of information between different regions with different regional languages. Machine Translation poses numerous challenges, some of which are: a) Not all words in one language has equivalent word in another language b) Two given languages may have completely different structures c) Words can have more than one meaning. Owing to these challenges, along with many others, MT has been active area of research for more than five decades. Numerous methods have been proposed in the past which either aim at improving the quality of the translations generated by them, or study the robustness of these systems by measuring their performance on many different languages. In this literature review, we discuss statistical approaches (in particular word-based and phrase-based) and neural approaches which have gained widespread prominence owing to their state-of-the-art results across multiple major languages.