Many engineering problems involve large material deformation, large boundary motion and continuous changes in boundary conditions. The Arbitrary Lagrangian-Eulerian (ALE) formulation has emerged in recent years as a technique that can alleviate many of the shortcomings of the traditional Lagrangian and Eulerian formulations in handling these types of problems. Using the ALE formulation the computational grid need not adhere to the material (Lagrangian) nor be fixed in space (Eulerian) but can be moved arbitrarily. Two distinct techniques are being used to implement the ALE formulation, namely the operator split approach and the fully coupled approach. A survey of the ALE literature shows that the majority of ALE implementations for quasi-static and dynamic analyses are based on the computationally convenient operator split technique. In addition, all previous dynamic ALE formulations are based on explicit time integration where no linearization is needed. This thesis presents a fully coupled implicit ALE formulation for the simulation of quasi-static large deformation and metal forming problems. ALE virtual work equations are derived from the basic principles of continuum mechanics. A method for the treatment of convective terms that sidesteps the computation of the spatial gradients of stresses is used in the derivation. The ALE virtual work equations are discretized using isoparametric finite elements. Full expression for the resulting ALE finite element matrices and vectors are given. A new relation that relates grid displacements with material displacements is introduced. A 2D finite element program, ALEFE, based on the presented formulation is developed and tested. The program may reduce to an updated Lagrangian or Eulerian methods as special cases. The output data format is designed to be compatible with general graphic simulation and data processing commercial softwares, so that contour x-y and deformed mesh plots may be easily created from the output data of ALEFE. The transfinite mapping method is used as the mesh motion scheme for internal nodes. A new treatment for mesh motion on material boundaries is introduced and implemented. Implicit time integration schemes are implemented in the code. Different numerical algorithms for the integration of the rate type constitutive equation are investigated and coupled with the return mapping algorithm to provide plastic incremental consistency. Jaumann and Truesdell rates are taken as the objective stress rates in the constitutive equation. Several quasi-static large deformation applications are solved using the developed code. Practical simulation cases include flat punch forging and sheet metal extrusion process. ALE results are in good agreement with Lagrangian results obtained by ABAQUS commercial FE software. ALE is shown to prevent mesh distortion and eliminate the need for special contact treatments for problems with frictionless contact. Keywords: ALE, Large Deformation Problem, Finite Element, Mesh Motion