We introduce a new numerical method, called "Surfer," for the simulation of two- and three-dimensional flows with several fluid phases and free interfaces between them. We consider incompressible fluids obeying the Navier-Stokes equation with Newtonian viscosity in the bulk of each phase. Capillary forces are taken into account even when interfaces merge or break up. Fluid interfaces are advanced in time using an exactly volume conserving variant of the volume of fluid algorithm, thus allowing for full symmetry between fluid phases. The Navier-Stokes equation is solved using staggered finite differences on a MAC grid and a split-explicit time differencing scheme, while incompressibility is enforced using an iterative multigrid Poisson solver. Capillary effects are represented as a stress tensor computed from gradients of the volume fraction function. This formulation is completely independent of the topology of interfaces and relatively easy to implement in (3 \mathrm{D}). It also allows exact momentum conservation in the discretized algorithm. Numerical spurious effects or "parasite currents" are noticed and compared to similar effects in Boltzmann lattice gas methods for immiscible fluids. Simulations of droplets pairs colliding in 2D and in 3D are shown. Interface reconnection is performed easily, despite the large value of capillary forces during reconnection. (c) 1994 Academic Press, Inc.