In recent years, the emerging topics of recommender systems that take advantage of natural language processing techniques have attracted much attention, and one of their applications is the Conversational Recommender System (CRS). Unlike traditional recommender systems with content-based and collaborative filtering approaches, CRS learns and models user's preferences through interactive dialogue conversations. In this work, we provide a summarization of the recent evolution of CRS, where deep learning approaches are applied to CRS and have produced fruitful results. We first analyze the research problems and present key challenges in the development of Deep Conversational Recommender Systems (DCRS), then present the current state of the field taken from the most recent researches, including the most common deep learning models that benefit DCRS. Finally, we discuss future directions for this vibrant area.