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

exchangedtheirmomenta.Oneeasilyseesthat(54)is

ful lled

for

an

arbitraryS(p2,p1)yieldingtheenergyasasumofone-particleenergies

E2=4sin2(p1

2).(57)

Eq.(55)thendeterminestheS-matrixtobeoftheform

S(p1,p2)= (p1) (p2)+ipcot(2

2 i<jθP(i)P(j) (61)

wherethesumisoverallM!permutationsofthelabels{1,2,...,M}andthephaseshiftsθij= θjiarerelatedtotheS-matrix(58)by

S(pi,pj)=exp[iθij].

TheM-magnonBetheansatzthenyieldsthesetofMBetheequations

eipkL=M (62)S(pk,pi)(63)

i=1,i=k