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

relatingtheintegrationconstantqtotheparametersofouransatz4.OurgoalistocomputetheenergyEofthesetwosolutionsasafunctionofthecommutingangularmomentaJ1andJ2onthethreespherewithinS5.Uponusingeqs.(7)andinsertingtheansatz(16)thesearegivenby

√E= 2π 2πdσdσλω1λω2

00

λ=J1

ω2.(23)

3.1Thespecialcase:ω1=ω2

Itisinstructiveto rstdiscusstheparticularlysimplespecialcaseofacircularstring,whereω1=ω2and(20)doesnotapply.Thiswillturnouttobealimitingcaseoftheq>1scenario.Forω21=0theequationofmotionforψ(σ)(18)immediatelyyields

ψ′′=0 ψ(σ)=nσ(24)

withnbeingtheintegerwindingnumberofthisciruclarstringψ(σ+2π)=ψ(σ)+√2πn.InthiscasetheVirasoroconstraints(6)yieldκ=

1+

J12n2λ!Thisamountstoanalllooppredictionforthedualgauge

theoryscalingdimensionintheBMNlimitJ1→∞withλ/J12 xed,quitesimilartotheresultfortheplane-wavesuperstringdiscussedabove.

Letusnowdiscussthefoldedandcircularstringsolutionsinturn.