Casimir force is a cause of stiction (adhesion) between metal surfaces in Micro-Electro Mechanical Systems (MEMS). Casimir Force depends strongly on the separation of the two surfaces and the contact area. This thesis reviews the theory and prior experimental demonstrations of the Casimir force. Then the Casimir attractive force is calculated for a particular MEMS cantilever device, in which the metal cantilever tip is required to repeatedly touch and release from a metal tip pad on the substrate surface in response to a periodic driving electrostatic force. The elastic force due to the bending of the cantilever support arms is also a consideration in the device operation. The three forces are calculated analytically and compared as a function of cantilever tip height. Calculation of the electrostatic force uses coefficients of capacitance and electrostatic induction determined numerically by the finite element method, including the effect of permittivity for the structural oxide. A condition on the tip area to allow electrostatic release of the tip from the surface against Casimir sticking and elastic restoring forces is established.