This dissertation describes the evolution of a method for accurately predicting unsteady, three-dimensional, viscous flows with free-surfaces. The special feature of this study is a time-accurate and divergence-free procedure for the computation of the free-surface motion. The method attempts to achieve second order temporal resolution of the free-surface motion through an iterative procedure. The momentum equations are also discretized using a time centered implicit differencing using information from two time levels. The formulation uses a surface-fitting approach which forces the free-surface to coincide with one boundary of the computational domain and solves the three-dimensional incompressible Navier-Stokes equations in primitive variable form within the liquid region. The algebraic equations resulting from a coupled implicit discretization of the governing equations are solved using a vectorized strongly implicit procedure that updates all the primitive variables (the pressure and the three components of velocity) in one calculation sweep. Several axisymmetric and truly three-dimensional cases in a variety of geometries and different levels of gravity have been computed. Results presented include verification of the transient results by comparing with experimental data for the broken dam problem which has served as a standard test case for other free-surface solvers;The procedure is also capable of resolving surface tension or microgravity effects through appropriate boundary conditions on the free surface. This capability is also verified as part of this study by calculating the motion of liquids inside partially filled cylinders. These include computations for widely differing levels of gravity as well as different initial conditions for the spin up from rest. Also included is the description of fluid-structure interaction; the fluid flow calculations are combined with a structural analysis code transferring information back and forth at each time step.