Abstract: Unbounded domain problems are frequently encountered in engineering. As a semi-analytical and semi-discretized numerical method, Finite Element Method of Lines (FEMOL) has shown good performance on this type of problems. Based on the proposed theory of infinite elements with mapping technique, the adaptive FEMOL with Element Energy Projection (EEP) super-convergent method is applied to the solution of 2D unbounded domain problems, in which users are only required to pre-specify an error tolerance and a rough initial mesh, and then an adaptive FEMOL mesh is automatically produced by the algorithm, on which the accuracy of FEMOL solution with both regular elements and infinite elements satisfies the specified error tolerance in maximum norm. An introduction of the theory of FEMOL and the infinite elements are given firstly, and then the strategy of adaptive FEMOL based on EEP method is presented. The feasibility of applying the adaptive FEMOL to unbounded domain problems is analyzed. Then three unbounded domain problems are adaptively solved, including the Poisson equation of flow around a circular cylinder, the plane problem of uniaxial tension of infinite plate with a circle hole in elasticity, and the semi-infinite half space body under uniformly distributed circular load. Finally the displacements (function solutions) satisfying the error tolerance can be obtained and the stresses (derivative solutions) with superior accuracy can be calculated. Therefore the adaptive FEMOL can be taken as a new approach for solution of unbounded domain problems.