Asymptotic properties of the solutions of SDEs have been widely studied in the past decades, particularly the stability theory has been attracting lots of attention. Due to the difficulty to find the explicit solutions to SDEs, different types of numerical methods have been introduced to approximate the underlying solutions. Thus the study of the stability of the numerical methods has naturally boomed in recent years. Another important asymptotic property of the SDE solutions, the asymptotic boundedness, has its own right. Unlike the stability property that requires the solutions be attracted by an equilibrium state, the boundedness property only requires the solutions stay within certain regime as time tends to infinity. Works on the boundedness of the underlying SDE solutions have been done by many authors. But there are few papers investigating the asymptotic boundedness of the numerical solutions. The main purpose of this talk is to investigate the asymptotic moment boundedness of two classical numerical methods. We focus on two types of moment, small moment (i.e. pth moment with p much smaller than 1) and second moment. Our key aim in this talk is to answer the question: given that the solution of the underlying Itô type SDE is asymptotically bounded in moment, is there any numerical method that could preserve the boundedness property? Due to the techniques used to deal with the small moment are much more complicated than those for the second moment, the majority of the talk is devoted to the case of small moment.X