This paper presents a homogenization method for shape and topology optimization of linearly elastic structures. The method, based on the work of Bendsoe and Kikuchi, introduces infinitely many microscale voids (holes) to form a possibly porous medium that yields a linearly elastic structure. The optimization problem is defined by solving the optimal porosity of the medium identified with a design domain. The method is applied to various example problems to justify its validity and strength for plane structures. The approach is extended to solve both shape and topology optimization problems using a fixed finite element model. The method is shown to be effective for solving layout problems in a generalized sense for any type of linearly elastic structures. The homogenized elasticity tensor is computed by solving the problem defined in the unit cell in which a rectangular hole is placed. The method is verified by solving a simple problem whose solution may be obtained analytically using a simple structural model such as a truss or a beam. The results show that the present method can not only solve shape optimization problems but also topology optimization problems for linearly elastic structures using a fixed finite element model. The convergence property of the finite element approximation is examined, and it is shown that the optimal configurations converge to the unique one as the finite element meshes are uniformly refined. The method is also shown to be able to reproduce the Michell truss, which is already known as the optimal truss structure for bending. The effect of the boundary condition is examined using a short beam used in the example for convergence study. The results show that the method is effective for solving shape and topology optimization problems for linearly elastic structures.This paper presents a homogenization method for shape and topology optimization of linearly elastic structures. The method, based on the work of Bendsoe and Kikuchi, introduces infinitely many microscale voids (holes) to form a possibly porous medium that yields a linearly elastic structure. The optimization problem is defined by solving the optimal porosity of the medium identified with a design domain. The method is applied to various example problems to justify its validity and strength for plane structures. The approach is extended to solve both shape and topology optimization problems using a fixed finite element model. The method is shown to be effective for solving layout problems in a generalized sense for any type of linearly elastic structures. The homogenized elasticity tensor is computed by solving the problem defined in the unit cell in which a rectangular hole is placed. The method is verified by solving a simple problem whose solution may be obtained analytically using a simple structural model such as a truss or a beam. The results show that the present method can not only solve shape optimization problems but also topology optimization problems for linearly elastic structures using a fixed finite element model. The convergence property of the finite element approximation is examined, and it is shown that the optimal configurations converge to the unique one as the finite element meshes are uniformly refined. The method is also shown to be able to reproduce the Michell truss, which is already known as the optimal truss structure for bending. The effect of the boundary condition is examined using a short beam used in the example for convergence study. The results show that the method is effective for solving shape and topology optimization problems for linearly elastic structures.