I explain a recent result on the noncompact Yamabe flow which is joint work with Jin Takahashi at Tokyo Institute of Technology. The noncompact Yamabe flow is complicated compared to the compact case. There are many unexpected phenomena from the viewpoint of the compact Yamabe flow. One of the remaining questions is the following. If each Riemannian metric is complete under the Yamabe flow on a noncompact manifold for all time and the long time limit exists, then is the limit also complete? I give the negative answer to this question by giving a counterexample.