A numerical method based on an adaptive octree space discretization for the simulation of displacement of free surfaces is proposed and applied to 3D free surface flow problems. A VOF approach is combined with a mass-conserving semi-Lagrangian time-stepping scheme. An interface prediction algorithm is used to refine the octree at the predicted location of the interface in order to ensure detail preservation. Subsequently, the fluid is advected and a coarsening algorithm adapts the mesh to avoid excess refinement in non-interfacial regions. SLIC and decompression algorithms are used for post-processing to limit numerical diffusion and correct numerical compression of the VOF function. The scheme is unconditionally stable with respect to the CFL number and does not require solving of a linear system. The octree scheme allows anisotropy and refinement of interfacial cells to an arbitrary level. It does not require a 2:1 cell size ratio condition between neighbouring cells. Numerical validation is done on benchmark test cases and results are compared with the structured analog. The scheme is coupled with a Stokes solver on a tetrahedral grid for solving of time-dependent Navier-Stokes equations and numerical results are compared with experimental water wave profile measurements. (C) 2016 The Authors. Published by Elsevier Ltd.