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.
