We present scalable algorithms for the level-set method on dynamic, adaptive Quadtree and Octree Cartesian grids. The algorithms are fully parallelized and implemented using the MPI standard and the open-source p4est library. We solve the level set equation with a semi-Lagrangian method which, similar to its serial implementation, is free of any time-step restrictions. This is achieved by introducing a scalable global interpolation scheme on adaptive tree-based grids. Moreover, we present a simple parallel reinitialization scheme using the pseudo-time transient formulation. Both parallel algorithms scale on the Stampede supercomputer, where we are currently using up to 4096 CPU cores, the limit of our current account. Finally, a relevant application of the algorithms is presented in modeling a crystallization phenomenon by solving a Stefan problem, illustrating a level of detail that would be impossible to achieve without a parallel adaptive strategy. We believe that the algorithms presented in this article will be of interest and useful to researchers working with the level-set framework and modeling multi-scale physics in general. (C) 2016 Elsevier Inc. All rights reserved.