In this talk I will present harmonic map flow and its variants. In the first part I introduce known behavior of harmonic map heat flow, including finite time bubbling. As its variation, I will explain Rupflin-Topping's Teichmuller flow. This flow is the L^2 gradient flow of the Dirichlet energy with respect to both the map and constant curvature metric over hyperbolic Riemann surface. Several results of Teichmuller flow will be covered. Finally, I will introduce a new variation of the harmonic map heat flow, called the conformal heat flow which is designed to delay finite time singularities.