  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, which
allowed to devise powerful methods ostensibly solving the spectral problem.
However, quantization of the AdS$_5\times\;$S$^5$ superstring from first
principles is still an open question and especially the spectrum of short
string states has previously been derived only at leading order in large 't
Hooft coupling. In this thesis we investigate possible routes to take the
perturbative quantization of short string states beyond the leading order,
where equally our aim is to gain better appreciation of the quantum symmetries
at play. A prominent role is played by the lowest excited string states, dual
to the Konishi supermultiplet, and we start by reviewing critically an asserted
derivation of the Konishi anomalous dimension in the setup of pure spinor
string theory. Next, we study the bosonic AdS$_5\times\;$S$^5$ string in static
gauge, where we construct a so-called single-mode string solution, which shows
classical integrability and invariance under the isometries
SO(2,4)$\;\times\;$SO(6) at the quantum level. Arguing heuristically about the
effects of supersymmetry, we indeed recover the first non-trivial quantum
correction to the Konishi anomalous dimension. We continue by implementing
static gauge for the full AdS$_5\times\;$S$^5$ superstring and find elegant
expressions for the Lagrangian density and the supercharges. We then constrain
our interest to the superparticle and, using two different methods, find
canonical coordinates at quadratic order in fermions. As the single-mode string
is just the SO(2,4)$\;\times\;$SO(6) orbit of the pulsating string, we conclude
by applying orbit method quantization to the particle and spinning string
solutions in bosonic AdS$_3\times\;$S$^3$ yielding consistent results for the
spectra.
