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The the duality between ${\cal N}=4$ SYM and the AdS$_5\times\;$S$^5$superstring appears to enjoy quantum integrability in the planar limit, whichallowed to devise powerful methods ostensibly solving the spectral problem.However, quantization of the AdS$_5\times\;$S$^5$ superstring from firstprinciples is still an open question and especially the spectrum of shortstring states has previously been derived only at leading order in large 'tHooft coupling. In this thesis we investigate possible routes to take theperturbative quantization of short string states beyond the leading order,where equally our aim is to gain better appreciation of the quantum symmetriesat play. A prominent role is played by the lowest excited string states, dualto the Konishi supermultiplet, and we start by reviewing critically an assertedderivation of the Konishi anomalous dimension in the setup of pure spinorstring theory. Next, we study the bosonic AdS$_5\times\;$S$^5$ string in staticgauge, where we construct a so-called single-mode string solution, which showsclassical integrability and invariance under the isometriesSO(2,4)$\;\times\;$SO(6) at the quantum level. Arguing heuristically about theeffects of supersymmetry, we indeed recover the first non-trivial quantumcorrection to the Konishi anomalous dimension. We continue by implementingstatic gauge for the full AdS$_5\times\;$S$^5$ superstring and find elegantexpressions for the Lagrangian density and the supercharges. We then constrainour interest to the superparticle and, using two different methods, findcanonical coordinates at quadratic order in fermions. As the single-mode stringis just the SO(2,4)$\;\times\;$SO(6) orbit of the pulsating string, we concludeby applying orbit method quantization to the particle and spinning stringsolutions in bosonic AdS$_3\times\;$S$^3$ yielding consistent results for thespectra.

Spectrum and Quantum Symmetries of the ${\rm AdS}_5 \times {\rm S}^5$ Superstring