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We analyze in detail the relation between an exactly marginal deformation ofN=4 SYM - the Leigh-Strassler or ``beta-deformation'' - and its string theorydual (recently constructed in hep-th/0502086) by comparing energies ofsemiclassical strings to anomalous dimensions of gauge-theory operators in thetwo-scalar sector. We stress the existence of integrable structures on the twosides of the duality. In particular, we argue that the integrability of stringsin AdS_5 x S^5 implies the integrability of the deformed world sheet theorywith real deformation parameter. We compare the fast string limit of theworldsheet action in the sector with two angular momenta with the continuumlimit of the coherent state action of an anisotropic XXZ spin chain describingthe one-loop anomalous dimensions of the corresponding operators and find aremarkable agreement for all values of the deformation parameter. We discusssome of the properties of the Bethe Ansatz for this spin chain, solve the Betheequations for small number of excitations and comment on higher loop propertiesof the dilatation operator. With the goal of going beyond the leading order inthe 't Hooft expansion we derive the analog of the Bethe equations on thestring-theory side, and show that they coincide with the thermodynamic limit ofthe Bethe equations for the spin chain. We also compute the 1/J corrections tothe anomalous dimensions of operators with large R-charge (corresponding tostrings with angular momentum J) and match them to the 1-loop corrections tothe fast string energies. Our results suggest that the impressive agreementbetween the gauge theory and semiclassical strings in AdS_5 x S^5 is part of alarger picture underlying the gauge/gravity duality.Comment: 50 pages, Latex; v2:typos corrected, references added, clarifications  in sec 8 and Appendix A, a discussion of a rational solution added in section  4.2; v3: minor corrections to coefficients in eq. 2.5, 5.2 and appendix A;  v4: minor misprints correcte

Gauge-string duality for superconformal deformations ofN= 4 Super Yang-Mills theory