In this thesis the rolling contact problem between two closely conforming elastic bodies are analyzed by the finite element method. The kinematic relation between two conforming bodies in rolling contact is derived. The kinematic relation and the stress contact conditions are transformed into multiple point constraints for nodal displacements. Upon imposing these constraints into the finite element system equations, the nodal displacements and stresses may satisfy all contact conditions. This technique may be used to obtain the contact length, and tangential as well as normal stresses in the contact region during static and rolling contact of two conforming bodies.