For many practical fluid flow problems, such as those found in I.C. engines and bio-devices, part or all of the boundaries are moving in space. An accurate and conservative representation of moving boundaries is essential for any numerical method. In this paper, a finite volume method is described to solve unsteady three-dimensional fluid flow problems with moving boundaries. The computational grid is allowed to move arbitrarily and to conform to the boundary motion. The approach could be categorized as an arbitrary Lagrangian-Eulerian method and has been designed for use on a general non-orthogonal grid. An extra geometric constraint called Space Conservation Law, is introduced and numerically implemented in an accurate and general way. This space conservation law is combined together with other conservation laws and the whole system is solved with a pressure-based numerical method on a non-staggered grid. Examples are given to demonstrate the importance of a fully conservative and accurate implementation of the space conservation law and the consequences of not satisfying it. The proposed approach has been implemented in order to solve 2D or 3D flows in a body-fitted-coordinate system. Practical usability of the method is emphasized. Three sample applications, two 2D and one 3D, are used to demonstrate the approach. Results are compared favorably to analytical, experimental, and other numerical solutions.