An　adaptive　finite　element　method　for　eigenvalue　problems　is　proposed　based　on　the　multilevel　correction　scheme.　Different　from　the　standard　adaptive　finite　element　method　which　requires　solving　eigenvalue　problems　on　adaptively　refined　triangulations,　our　scheme　only　includes　solving　associated　boundary　value　problems　on　adaptive　triangulations　and　small　scale　eigenvalue　problems　on　a　very　low　dimensional　space.　Since　there　is　no　eigenvalue　problem　to　be　solved　on　adaptively　refined　triangulations,　which　needs　more　computation　and　memory　than　solving　associated　boundary　value　problem,　the　efficiency　of　the　proposed　method　can　be　improved　to　be　similar　to　that　of　the　adaptive　finite　element　method　for　associated　boundary　value　problems.　The　convergence　and　optimal　complexity　is　theoretically　verified　and　numerically　demonstrated.