This　paper　documents　the　first　attempt　to　apply　the　generalized　finite　difference　method　(GFDM),　a　recently　developed　meshless　method,　for　the　numerical　solution　of　problems　with　cracks　in　general　anisotropic　materials.　To　solve　the　resulting　second-order　elliptic　partial　differential　equations　with　mixed　boundary　conditions,　the　explicit　formulae　for　the　partial　derivatives　of　unknown　functions　in　the　equations　are　derived　by　using　the　Taylor　series　expansions　combining　with　the　moving-least　squares　approximation　in　this　meshless　GFD　method.　To　deal　with　the　strong　discontinuous　crack-faces,　some　special　treatments　are　applied　to　modify　this　GFDM.　The　node　distributions　are　locally　refined　in　the　vicinity　of　the　crack-tips.　By　dividing　the　crack　domain　into　two　parts,　the　sub-domain　method　is　also　used　for　comparing　with　the　single-domain　method.　The　direct　displacement　extrapolation　method,　the　path-independent　J-integral　and　the　interaction　integral　methods　are,　respectively,　used　to　compute　the　stress　intensity　factors　and　compared.　Finally,　some　classical　crack　examples　are　presented　to　show　the　effectiveness　and　accuracy　of　the　proposed　meshless　method　for　crack　problems.