Inrecentyearstherehasbeenenormousactivityinthetheoryofalgebraiccurves．Manylong・standingproblemshavebeensolvedusingthegeneraltechniquesdevelopedinalgebraicgeometryduringthe1950’Sand1960’S．Additionally，unexpectedanddeepconnectionsbetweenalgebraiccurvesanddifferentialequationshavebeenuncovered．andtheseinturnshedlightonotherclassicalproblemsincurvetheory．Itseemsfairtosaythatthetheoryofalgebraiccurveslookscompletelyditierentnowfromhowitappeared15yearsago；inparticular，ourcurrentstateofknowledgerepre-sentsasignificantadvancebeyondthelegacyleftbytheclassicalgeometerssuchasNoether，Castelnuovo，Enriques，andSeveri．

　　Thesebooksgiveapresentationofoneofthecentralareasofthisrecentactivity：namely，thestudyoflinearseriesonbothafixedcurve（VolumeI）andonavariablecurverVolumeII）．OurgoaliStogiveacomprehensiveandself-containedaccountoftheextrinsicgeometryofalgebraiccurves，whichinouropinionconstitutesthemaingeometriccoreoftherecentadvancesincurvetheory．Alongthewayweshall，ofcourse，discussappli．cationsofthetheoryoflinearseriestoanumberofclassicaltopics（e…gthegeometryoftheRiemannthetadivisor）aswellastosomeofthecurrentresearchr（e.gtheKodairadimensionofthemodulispaceofcurves）．