In this talk, I will present our recent analysis results on some car-following models, i.e., the intelligent driver's model (IDM) and the Bando-Follow the leader (Bando-FtL) model. In particular, we show the limitations of the IDM and propose several modifications to the IDM in order to avoid such ill-posedness. For the Bando-FtL model, under suitable conditions, we provide existence and uniqueness results, and also bounds on the higher derivatives, i.e. velocity and acceleration. We then extend the result to the ``reaction'' delay case where the delay is instantiated in reacting to the leading vehicle's position and velocity. We prove that the solution of the delayed model converges to the undelayed when the delay converges to zero. Some numerical results will also be presented.