Many important real-world problems can be described by systems of ODEs whose solution possesses some strong boundedness property. Such systems frequently arise from semi-descretizations of PDEs. Examples of such boundedness properties include monotonicity, contractivity, and positivity. I will discuss the preservation of such stability properties under discretization using Runge-Kutta and linear multistep methods. For methods of order greater than one, stability may be preserved only for timesteps smaller than some value that is porportional to a factor depending purely on the choice of ODE solver. This leades us to seek methods and discuss my own current efforts (in collaboration with Sigal Gottlieb and Colin MacDonald) in this area.