Anisotropic　mesh　adaptation　is　studied　for　the　linear　finite　element　solution　of　eigenvalue　problems　with　anisotropic　diffusion　operators.　The　M-uniform　mesh　approach　is　employed　with　which　any　nonuniform　mesh　is　characterized　mathematically　as　a　uniform　one　in　the　metric　specified　by　a　metric　tensor.　Bounds　on　the　error　in　the　computed　eigenvalues　are　established　for　quasi-M-uniform　meshes.　Numerical　examples　arising　from　the　Laplace.Beltrami　operator　on　parameterized　surfaces　and　nonlinear　diffusion　problems　in　image　processing　and　radiation　hydrodynamics　are　presented.　Numerical　results　show　that　anisotropic　adaptive　meshes　can　lead　to　more　accurate　computed　eigenvalues　than　uniform　or　isotropic　adaptive　meshes.　They　also　confirm　the　second　order　convergence　of　the　error　that　is　predicted　by　the　theoretical　analysis.　The　effects　of　approximation　of　curved　boundaries　on　the　computation　of　eigenvalue　problems　is　also　studied　in　two　dimensions.　It　is　shown　that　the　initial　mesh　used　to　define　the　geometry　of　the　physical　domain　should　contain　at　least　√N　boundary　points　to　keep　the　effects　of　boundary　approximation　at　the　level　of　the　error　of　the　finite　element　approximation,　where　N　is　the　number　of　the　elements　in　the　final　adaptive　mesh.　Only　about　3√N　boundary　points　in　the　initial　mesh　are　needed　for　boundary　value　problems.　This　implies　that　the　computation　of　eigenvalue　problems　is　more　sensitive　to　the　boundary　approximation　than　that　of　boundary　value　problems.　©　2015　Society　for　Industrial　and　Applied　Mathematics.