Several phenomena in the physical and the life sciences can be modeled as time dependent free boundary problems and nonlinear partial differential equations. Examples include the study of electro-osmotic flows, solidification of binary alloys, free surface flows and multiphase flows in porous media. One of the main difficulties in solving numerically these equations is associated with the fact that such problems involve dissimilar length scales, with smaller scales influencing larger ones so that nontrivial pattern formation dynamics can be expected to occur at all intermediate scales. Uniform grids are limited in their ability to resolve small scales and are in such situations extremely inefficient in terms of memory storage and CPU requirements. Another difficulty stems from the fact that the geometry of the problems is often arbitrary and special care is needed to correctly apply boundary conditions. In this talk, I will present recent advances in the numerical treatment of interface problem and describe new numerical solvers for nonlinear partial differential equations in the context of adaptive mesh refinement based on Octree grids. If time permits, I will also present a method for accurately simulating fluid-solid two-way coupling.