In this talk I will discuss the contrasting capabilities of lattice- and off-lattice kinetic Monte Carlo simulation, illustrating each with a particular application. For lattice based simulation I will consider the role of surface energy anisotropy in stabilizing the Bales-Zangwill meandering instability of step-flow. I will also briefly discuss a simple model for graphene and the novel way surface energy behaves in that system. For the off-lattice case I will introduce a rejection-based algorithm that offers the potential to greatly accelerate such simulations. This will be demonstrated by applying the method to the growth of two-species nanoclusters with a core-shell structure.