In 1964, Eells and Sampson asked whether  a given smooth map   can be deformed to a harmonic map  in its homotopy class . When $n=2$, Lemaire   and Schoen-Yau established  existence results of harmonic maps by minimizing the Dirichet energy  in a homotopy class under the  topological condition $\pi_2(N)=0$.   Sacks and Uhlenbeck established many existence results  of minimizing harmonic maps in their homotopy classes by introducing the `Sacks-Uhlenbeck functional'. To solve the   Eells-Sampson question,  it is important to establish  global existence of a solution of the harmonic map flow. Struwe proved  the existence of a unique global weak solution  to the harmonic map flow.   Chang, Ding and Ye constructed an example that the harmonic map flow  blows up at finite time.  Ding and Tian established the energy identity of the harmonic map flow at each blow-up time.  

When $n>2$, we also introduce some new result on a  $n$-harmonic map flow from an n-dimensional closed Riemannian manifold to another closed  Riemannian manifold.

Finally, we mention some applications of the n-harmonic map flow to minimizing the n-energy functional and the Dirichlet energy functional in a homotopy class.