In this talk, we discuss the theory of (minimizing)  harmonic and p-harmonic mappings between Riemannian manifolds  as well as singular metric spaces. I will start my talk with  an overview of the existence, uniquness and regularity of  harmonic mappings between Riemannian manifolds. Then I will  point out the corresponding extension of the theory to p- harmonic mappings in a similar setting. In the final part, I  will discuss the development of the harmonic mapping theory  (in the sense of Korevaar-Schoen) in the setting of singular  metric measure spaces. A couple of open problems in the  project of the speaker for future research will be discussed  at the end of the talk.