Finite element analysis of fluid flow with moving free surface has been carried out in two and tree dimensions. The new VOF-based numerical algorithm that has been proposed by the present authors was applied to several 2-D and 3-D free surface flow problems. The proposed free surface tracking scheme is based on two numerical tools that have been newly introduced by the present authots; the orientation vector to represent the free surface orientation in each cell and the baby-cell to determine the fluid volume flux at each cell boundary. The proposed numerical algorithm has been applied to 2-D and 3-D cavity filling and sloshing problems, which demonstrated versatility and effectiveness of the new free surface tracking scheme as well as the overall solution procedure. The proposed numerical algorithm resolved successfully the interacting free surface with each other. The simulated results demonstrated the applicability of proposed numerical algorithm to the practical problems of large free surface motion. Also, it has been demonstrated that the proposed free surface tracking scheme can be easily implemented in any irregular non-uniform grid systems and can be extended to the 3-D free surface flow problem without additional efforts.