In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the

aredescribedbyclassicalsupergravityandweaklycoupledgauge eldscorrespondtostringspropagatinginahighlycurvedbackgroundgeometry.Thisinsightiscertainlyfascinating,butatthesametimestronglyhindersanydynamicaltests(orevenaproof)oftheAdS/CFTconjectureinregimeswhicharenotprotectedbythelargeamountofsymmetryintheproblem.

Thissituationhasprofoundlychangedsince2002byperformingstudiesofthecorrespondenceinnovellimitswherequantumnumbers(suchasspinsorangularmomentainthegeometricAdS5

N→∞.Thiswasinitiated×inS5language)becomelargeinacorrelatedfashionastheworkofBerenstein,MaldacenaandNastase

[5]whoconsideredthequantum uctuationexpansionofthestringaroundade-generatedpoint-likecon guration,correspondingtoaparticlerotatingwithalargeangularmomentumJonagreatcircleoftheS5space.InthelimitofJ→∞withJ2/Nheld xed(the‘BMNlimit’)thegeometryseenbythefastlymovingparticleisagravitationalplane-wave,whichallowsforanexactquantizationofthefreestringinthelight-cone-gauge[6,7].Theresultingstringspectrumleadstoaformidablepredictionfortheallloopscalingdimensionsofthedualgauge

inthecorrespondinglimit,i.e.thefamousformula n=J+2 theoryoperatorsJ2forthe

simplesttwostringoscillatormodeexcitation.Thekeypointhereistheemergenceofthee ectivegaugetheoryloopcountingparameterλ/J2intheBMNlimit.Thesescalingdimensionshavebynowbeen rmlyreproduceduptothethreelooporderingaugetheory[8,9,10].Thishasalsoledtoimportantstructuralinformationforhigher(orallloop)attemptsingaugetheory,whichmaximallyemploytheuncov-eredintegrablestructurestobediscussedbelow.Moreover,theplanewavestringtheory/N=4SYMdualitycouldbeextendedtotheinteractingstring(gS

respectivelynon-planargaugetheoryregimeprovidinguswiththemostconcrete=0)realizationofastring/gaugedualitytodate(forreviewssee[11,12,13,14]).

InthisreviewweshalldiscussdevelopmentsbeyondtheBMNplane-wavecor-respondencewhichemploymoregeneralsectorsoflargequantumnumbersintheAdS/CFTduality.Thekeypointfromthestringperspectiveisthatsuchalimitcanmakethesemiclassical(intheplane-wavecase)orevenclassical(inthe‘spin-ningstring’case)computationofthestringenergiesalsoquantumexact[15,16],i.e.higherσ-modelloopsaresuppressedbyinversepowersofthetotalangularmomentumJonthe vesphere2.Theseconsiderationsonthestringsidethen(arguably)yieldalllooppredictionsforthedualgaugetheory.Additionallytheperturbativegaugetheoreticstudiesatthe rstfewordersinλledtothediscoverythatthespectrumofscalingdimensionsoftheplanargaugetheoryisidenticaltothatofanintegrablelong-rangespinchain[19,8,20].ConsistentlytheAdS5

stringisaclassicallyintegrablemodel[21],whichhasbeenheavilyexploitedin×theS5