In this talk,we discuss the blow-up behavior of delay differential equations (DDEs) with two kinds of vanishing delays,proportional delay and piecewise constant delay.Some fundamental results on the local existence and uniqueness of solutions to DDEs are reviewed and some conditions are presented under which the unique solution exists globally.We first offer some examples to illustrate that the blow-up behaviors of corresponding ODEs may be influenced by vanishing delays.Based on the monotonicity,we investigate conditions yielding a blow-up solution for the two kinds of DDEs,respectively.In the last part,we provided an adaptive stepsize strategy to simulate the blow-up behaviors of DDEs with proportional delay by collocation methods.