While　many　experiments　and　simulations　on　metallic　glasses　(MGs)　have　focused　on　their　tensile　ductility　under　monotonic　loading,　the　fatigue　mechanisms　of　MGs　under　cyclic　loading　still　remain　largely　elusive.　Here　we　perform　molecular　dynamics　(MD)　and　finite　element　simulations　of　tension-compression　fatigue　tests　in　MGs　to　elucidate　their　fatigue　mechanisms　with　focus　on　the　sample　size　effect.　Shear　band　(SB)　thickening　is　found　to　be　the　inherent　fatigue　mechanism　for　nanoscale　MGs.　The　difference　in　fatigue　mechanisms　between　macroscopic　and　nanoscale　MGs　originates　from　whether　the　SB　forms　partially　or　fully　through　the　cross-section　of　the　specimen.　Furthermore,　a　qualitative　investigation　of　the　sample　size　effect　suggests　that　small　sample　size　increases　the　fatigue　life　while　large　sample　size　promotes　cyclic　softening　and　necking.　Our　observations　on　the　size-dependent　fatigue　behavior　can　be　rationalized　by　the　Gurson　model　and　the　concept　of　surface　tension　of　the　nanovoids.　The　present　study　sheds　light　on　the　fatigue　mechanisms　of　MGs　and　can　be　useful　in　interpreting　previous　experimental　results.　(C)　2017　Elsevier　Ltd.　All　rights　reserved.