I will discuss ongoing research on the development of novel methods for the temporal integration of ODEs and PDEs. The strategy employed in the numerical methods is based on an iterative deferred corrections approach, within which we can utilize operator splitting, implicit-explicit methods, and parallelization in the temporal direction to achieve better computational efficiency. Much of the recent work is motivated by fluid-structure interaction problems in biological systems and I will discuss the difficulties and progress related to some target applications in this area.