In this paper, the concept of a new φ-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from φ-generalized quasi metric space. Some general fixed point theorems in a φ-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.