Style is an integral component of a sentence indicated by the choice of words a person makes. Different people have different ways of expressing themselves, however, they adjust their speaking and writing style to a social context, an audience, an interlocutor or the formality of an occasion. Text style transfer is defined as a task of adapting and/or changing the stylistic manner in which a sentence is written, while preserving the meaning of the original sentence. A systematic review of text style transfer methodologies using deep learning is presented in this paper. We point out the technological advances in deep neural networks that have been the driving force behind current successes in the fields of natural language understanding and generation. The review is structured around two key stages in the text style transfer process, namely, representation learning and sentence generation in a new style. The discussion highlights the commonalities and differences between proposed solutions as well as challenges and opportunities that are expected to direct and foster further research in the field.