Conformal Heat Flow is a pair of evolution equations of a map and a metric on the domain. This new type of flow can be understood as a harmonic map flow with metric evolution which is in conformal direction. In this talk, I will present basic idea of conformal heat flow of harmonic maps in 2-dimensional domain where the metric evolves almost proportional to the energy density. As its variant, I also introduce the system with Yamabe flow, which is in higher dimensions and the conformal factor satisfies a kind of Yamabe flow. This is a joint work with Hyo Seok Jang and Ki-Ahm Lee.