In　this　paper,　the　general　equations　of　plane　strain　problems　are　first　summarized　for　three　typical　couple　stress　theories.　Based　on　the　constitutive　relations,　the　general　traction　boundary　integral　equations　(TBIEs)　are　then　derived　from　the　displacement　BIEs　(DBIEs)　along　with　the　explicit　high-order　fundamental　solutions　for　the　plane　couple-stress　problems.　Then,　the　boundary　element　method　(BEM)　codes　based　on　the　DBIEs　(DBEM)　and　TBIEs　(TBEM)　discretized　by　the　collocation　scheme　are　developed　separately.　Some　typical　elastic　couple-stress　problems　are　numerically　studied.　For　crack　problems,　the　displacement　extrapolation　formulas　are　derived　for　computing　the　stress　and　couple-stress　intensity　factors.　Good　agreements　are　observed　between　the　results　obtained　by　the　DBEM　and　the　TBEM.　The　high　accuracy　of　the　present　BEM　results　is　verified　by　comparing　them　with　the　analytical　results.