The theory of order-preserving dynamical systems was largely developed in 1980's and 90's after the pioneering work of M.W. Hirsch and others. What is remarkable about this theory is that it allows us to derive various important qualitative properties of solutions -- such as stability and convergence -- solely by a slightly stronger version of the usual comparison principle, without further knowledge of the specific features of the equations. Recently there have been some new develpments in this theory. In this talk I will present our new results on order-preserving systems with a mass conservation property (or a first integral). Our results extend the earlier work by Arino (1991), Mierczynski (1995, 2012) and Banaji-Angeli (2010) considerably with a significantly simpler proof. I will then apply this theory to a number of problems including mathematical models for transportation by molecular motors, chemical reservible reactions, competition-diffusion systems, and so on. This is joint work with Toshiko Ogiwara and Danielle Hilhorst.